Asymptotic freeness of representation images of the canonical matrix

Let $\lambda^1(N),\lambda^2(N)\in\widehat U(N)$, for $N=1,2,\dots$, be two sequences of signatures satisfying the technical assumption of Definition $\mathrm{regularity}$, and suppose that

$$\lim_{N\to\infty}m_{PP}[\lambda^i(N)]=\mathbf m^i,\quad i=1,2.$$

For each $N$, let $\mathbf A(N)$ be the corresponding non-commutative probability space, and let $E(\lambda^1(N))$ and $E(\lambda^2(N))$ be the representation images of the canonical matrix element. Asymptotic freeness conjecture. As $N\to\infty$, the elements

$$\frac{1}{N}E(\lambda^1(N))\quad\text{and}\quad\frac{1}{N}E(\lambda^2(N))$$

become asymptotically free. This conjecture predicts that matrix images arising from two independent representation factors exhibit asymptotic freeness, complementing the preceding result that their sum has a limiting distribution given by the free convolution of the corresponding limiting measures. The statement is presented as a conjecture in the source, and no resolution is supplied here.