Let $(A,\varphi)$ be a unital noncommutative probability space over $\mathbb{C}$, with unit $1_A\in A$ and unital linear functional $\varphi:A\to\mathbb{C}$. For each integer $n\geq 1$, let $\kappa_n:A^n\to\mathbb{C}$ be the $n$-linear free cumulant functional determined by the moment-cumulant formula over noncrossing partitions. For every integer $n\geq 2$, every $x_1,\dots,x_n\in A$, and every index $j\in\{1,\dots,n\}$, if $x_j=1_A$, then $\kappa_n(x_1,\dots,x_n)=0$.