Let $(X,\mathcal B,\mu,T)$ be a Bernoulli shift over a finite alphabet $A$, where $X=A^{\mathbb Z}$ and $\mu=p^{\mathbb Z}$ for a probability vector $p$ on $A$. Let $\mathcal P$ be the coordinate partition according to the zeroth coordinate. Then $\mathcal P$ is very weak Bernoulli: for every $\varepsilon>0$, for all block lengths $n\in\mathbb N$ and sufficiently large gaps $g$, outside a past set of measure less than $\varepsilon$, the conditional distribution of the future-name map

\begin{align*} \Phi_n:X\to A^n,\quad \Phi_n(x)=(x_0,x_1,\dots,x_{n-1}) \end{align*}

is within Ornstein $\bar d_n$-distance less than $\varepsilon$ of its unconditional distribution.