[proofplan]
We fix a mixed word $x_1 \cdots x_n$ and encode the subalgebra membership of its letters by the colour map $j \mapsto i_j$. The free moment-cumulant formula expands $\varphi(x_1 \cdots x_n)$ as a finite sum over noncrossing partitions, with each summand a product of cumulants over the blocks of the partition. Freeness is equivalent to the vanishing of every free cumulant whose entries come from more than one of the freely independent subalgebras, so only colour-homogeneous blocks can contribute. On every surviving block, the cumulant is one of the given marginal cumulants, and hence the whole finite sum is uniquely determined.
[/proofplan]
[step:Expand the mixed moment by the free moment-cumulant formula]
Fix $n \in \mathbb{N}$, indices $i_1,\dots,i_n \in I$, and elements $x_j \in A_{i_j}$ for $1 \leq j \leq n$. Let $[n] := \{1,\dots,n\}$, and let $NC(n)$ denote the finite set of noncrossing partitions of $[n]$.
For a block $V \subset [n]$, write its elements in increasing order as $V = \{v_1 < \cdots < v_r\}$, where $r = |V|$. Define the block cumulant map $\kappa_V: A^n \to \mathbb{C}$ by
\begin{align*}
\kappa_V(a_1,\dots,a_n) := \kappa_r(a_{v_1},\dots,a_{v_r})
\end{align*}
for every $(a_1,\dots,a_n) \in A^n$. For a partition $\pi \in NC(n)$, define the partition cumulant map $\kappa_\pi: A^n \to \mathbb{C}$ by
\begin{align*}
\kappa_\pi(a_1,\dots,a_n) := \prod_{V \in \pi} \kappa_V(a_1,\dots,a_n)
\end{align*}
for every $(a_1,\dots,a_n) \in A^n$.
By the free moment-cumulant formula, applied to the noncommutative probability space $(A,\varphi)$ and the elements $x_1,\dots,x_n \in A$, we have
\begin{align*}
\varphi(x_1 \cdots x_n) = \sum_{\pi \in NC(n)} \kappa_\pi(x_1,\dots,x_n).
\end{align*}
[guided]
We begin by fixing the particular mixed moment we want to determine. Thus $n \in \mathbb{N}$ is fixed, $i_1,\dots,i_n \in I$ are fixed, and for each $1 \leq j \leq n$ we have an element $x_j \in A_{i_j}$. The index $i_j$ records the subalgebra from which the $j$-th letter of the word is being regarded as coming.
The free moment-cumulant formula expresses the moment of a word in terms of free cumulants indexed by noncrossing partitions. Let $[n] := \{1,\dots,n\}$, and let $NC(n)$ be the finite set of noncrossing partitions of $[n]$. If $V \subset [n]$ is a block, write its elements in increasing order as $V = \{v_1 < \cdots < v_r\}$, where $r = |V|$. The cumulant attached to this block is the map $\kappa_V: A^n \to \mathbb{C}$ defined by
\begin{align*}
\kappa_V(a_1,\dots,a_n) := \kappa_r(a_{v_1},\dots,a_{v_r})
\end{align*}
for every $(a_1,\dots,a_n) \in A^n$. This definition is needed because a block of a partition selects only some of the letters of the word, and the order of those selected letters is the order inherited from the original word.
For a noncrossing partition $\pi \in NC(n)$, define the partition cumulant map $\kappa_\pi: A^n \to \mathbb{C}$ by multiplying the block cumulants:
\begin{align*}
\kappa_\pi(a_1,\dots,a_n) := \prod_{V \in \pi} \kappa_V(a_1,\dots,a_n)
\end{align*}
for every $(a_1,\dots,a_n) \in A^n$. Each factor is a complex number, so the product is a product in $\mathbb{C}$.
Now apply the free moment-cumulant formula. Its hypotheses are satisfied because $(A,\varphi)$ is a noncommutative probability space and $x_1,\dots,x_n$ are elements of $A$. The formula gives
\begin{align*}
\varphi(x_1 \cdots x_n) = \sum_{\pi \in NC(n)} \kappa_\pi(x_1,\dots,x_n).
\end{align*}
Thus the mixed moment will be determined once we know which summands in this finite expansion survive and how to evaluate them from the marginal cumulant data.
[/guided]
[/step]
[step:Use freeness to remove partitions with mixed-colour blocks]
For a block $V \subset [n]$, call $V$ colour-homogeneous if there exists $i \in I$ such that $i_j = i$ for every $j \in V$. Call a partition $\pi \in NC(n)$ colour-homogeneous if every block $V \in \pi$ is colour-homogeneous.
Suppose $\pi \in NC(n)$ is not colour-homogeneous. Then there exists a block $V = \{v_1 < \cdots < v_r\} \in \pi$ and two positions $v_a,v_b \in V$ such that $i_{v_a} \neq i_{v_b}$. Hence the entries $x_{v_1},\dots,x_{v_r}$ of the cumulant $\kappa_r(x_{v_1},\dots,x_{v_r})$ come from at least two distinct subalgebras among the freely independent family $(A_i)_{i \in I}$. By the mixed-cumulant vanishing hypothesis stated in the theorem, this block cumulant vanishes:
\begin{align*}
\kappa_V(x_1,\dots,x_n) = \kappa_r(x_{v_1},\dots,x_{v_r}) = 0.
\end{align*}
Therefore
\begin{align*}
\kappa_\pi(x_1,\dots,x_n) = 0.
\end{align*}
Consequently the moment-cumulant expansion reduces to the sum over colour-homogeneous noncrossing partitions:
\begin{align*}
\varphi(x_1 \cdots x_n) = \sum_{\substack{\pi \in NC(n), \pi \text{ colour-homogeneous}}} \kappa_\pi(x_1,\dots,x_n).
\end{align*}
[guided]
The colour map $j \mapsto i_j$ records which subalgebra supplies each letter $x_j$. For a block $V \subset [n]$, call $V$ colour-homogeneous if there exists $i \in I$ such that $i_j = i$ for every $j \in V$. A partition $\pi \in NC(n)$ is colour-homogeneous if every one of its blocks has this property.
Now suppose $\pi \in NC(n)$ is not colour-homogeneous. Then at least one block $V = \{v_1 < \cdots < v_r\} \in \pi$ contains two positions $v_a$ and $v_b$ with $i_{v_a} \neq i_{v_b}$. Therefore the entries $x_{v_1},\dots,x_{v_r}$ in the cumulant $\kappa_r(x_{v_1},\dots,x_{v_r})$ do not all lie in one member of the freely independent family $(A_i)_{i \in I}$.
The theorem statement includes the mixed-cumulant vanishing form of freeness: a free cumulant whose arguments come from at least two distinct members of the freely independent family is zero. Applying that hypothesis to the tuple $(x_{v_1},\dots,x_{v_r}) \in A^r$ gives
\begin{align*}
\kappa_V(x_1,\dots,x_n) = \kappa_r(x_{v_1},\dots,x_{v_r}) = 0.
\end{align*}
Since $\kappa_\pi(x_1,\dots,x_n)$ is the product of the block cumulants over $V \in \pi$, this single zero factor forces
\begin{align*}
\kappa_\pi(x_1,\dots,x_n) = 0.
\end{align*}
Thus every non-colour-homogeneous partition contributes zero to the free moment-cumulant expansion. The only possible surviving summands are those indexed by colour-homogeneous noncrossing partitions, so
\begin{align*}
\varphi(x_1 \cdots x_n) = \sum_{\substack{\pi \in NC(n), \pi \text{ colour-homogeneous}}} \kappa_\pi(x_1,\dots,x_n).
\end{align*}
[/guided]
[/step]
[step:Evaluate every surviving block from the marginal cumulant data]
Let $\pi \in NC(n)$ be colour-homogeneous, and let $V = \{v_1 < \cdots < v_r\}$ be a block of $\pi$. Since $V$ is colour-homogeneous, there exists $i(V) \in I$ such that $i_{v_\ell} = i(V)$ for every $1 \leq \ell \leq r$. Hence
\begin{align*}
x_{v_1},\dots,x_{v_r} \in A_{i(V)}.
\end{align*}
The corresponding block cumulant is therefore
\begin{align*}
\kappa_V(x_1,\dots,x_n) = \kappa_r|_{A_{i(V)}^r}(x_{v_1},\dots,x_{v_r}).
\end{align*}
By hypothesis, the restricted cumulant functional $\kappa_r|_{A_{i(V)}^r}: A_{i(V)}^r \to \mathbb{C}$ is known. Thus every block cumulant in $\kappa_\pi(x_1,\dots,x_n)$ is known, and therefore the product
\begin{align*}
\kappa_\pi(x_1,\dots,x_n) = \prod_{V \in \pi} \kappa_V(x_1,\dots,x_n)
\end{align*}
is known.
[guided]
Fix a colour-homogeneous partition $\pi \in NC(n)$ and a block $V = \{v_1 < \cdots < v_r\}$ of $\pi$. Because $V$ is colour-homogeneous, there is an index $i(V) \in I$ such that every selected position in $V$ has colour $i(V)$; that is, $i_{v_\ell} = i(V)$ for every $1 \leq \ell \leq r$. Since the original word satisfies $x_j \in A_{i_j}$ for every $j$, this gives
\begin{align*}
x_{v_1},\dots,x_{v_r} \in A_{i(V)}.
\end{align*}
The block cumulant attached to $V$ is therefore not a genuinely mixed cumulant. It is exactly the restriction of the $r$-th cumulant to the subalgebra $A_{i(V)}$ evaluated at the selected tuple:
\begin{align*}
\kappa_V(x_1,\dots,x_n) = \kappa_r|_{A_{i(V)}^r}(x_{v_1},\dots,x_{v_r}).
\end{align*}
By the hypothesis of the theorem, the map $\kappa_r|_{A_{i(V)}^r}: A_{i(V)}^r \to \mathbb{C}$ is part of the known marginal cumulant data. Hence this block cumulant is known.
The partition cumulant is the finite product of these known block cumulants:
\begin{align*}
\kappa_\pi(x_1,\dots,x_n) = \prod_{V \in \pi} \kappa_V(x_1,\dots,x_n).
\end{align*}
A finite product in $\mathbb{C}$ of known complex numbers is known, so every surviving colour-homogeneous summand is determined by the marginal cumulant data.
[/guided]
[/step]
[step:Conclude that the finite surviving sum uniquely determines the mixed moment]
The set $NC(n)$ is finite, so the colour-homogeneous subset of $NC(n)$ is finite. The preceding step shows that each summand
\begin{align*}
\kappa_\pi(x_1,\dots,x_n)
\end{align*}
appearing in
\begin{align*}
\varphi(x_1 \cdots x_n) = \sum_{\substack{\pi \in NC(n), \pi \text{ colour-homogeneous}}} \kappa_\pi(x_1,\dots,x_n)
\end{align*}
is determined by the given family of restricted cumulant functionals $\kappa_m|_{A_i^m}: A_i^m \to \mathbb{C}$. Therefore the finite sum is uniquely determined by the marginal cumulant data. Since $n$, the indices $i_1,\dots,i_n$, and the elements $x_j \in A_{i_j}$ were arbitrary, every mixed moment of variables from $\bigcup_{i \in I} A_i$ is uniquely determined.
[guided]
The reduction has rewritten the desired moment as
\begin{align*}
\varphi(x_1 \cdots x_n) = \sum_{\substack{\pi \in NC(n), \pi \text{ colour-homogeneous}}} \kappa_\pi(x_1,\dots,x_n).
\end{align*}
The indexing set is finite because $NC(n)$ is finite and the colour-homogeneous partitions form a subset of it. For each partition appearing in this finite sum, the previous step proved that $\kappa_\pi(x_1,\dots,x_n)$ is determined by the known maps $\kappa_m|_{A_i^m}: A_i^m \to \mathbb{C}$.
A finite sum of determined complex numbers is determined. Hence the value of $\varphi(x_1 \cdots x_n)$ is uniquely determined by the marginal cumulant data. The initial choices of $n \in \mathbb{N}$, indices $i_1,\dots,i_n \in I$, and elements $x_j \in A_{i_j}$ were arbitrary, so the same argument applies to every mixed word in variables from $\bigcup_{i \in I} A_i$.
[/guided]
[/step]
