The free multiplicative convolution conjecture for graph wreath products

Let $X$ and $Y$ be colored graphs. Write $X*Y$ for the colored graph obtained by putting a copy of $X$ at each vertex of $Y$, let $\mu(Z)$ denote the spectral measure of the character of the algebra $A(Z)$ associated to a graph $Z$, and let $\boxtimes$ denote Voiculescu's free multiplicative convolution. Free multiplicative convolution conjecture. The spectral measures satisfy

$$\mu(X*Y)=\mu(X)\boxtimes\mu(Y).$$

This conjecture concerns the behavior of character spectral measures under the graph operation $X*Y$ and is presented as evidence-based work on analogues of spectral-measure formulae for quantum permutation groups; its resolution is not given in the source.