Brown measure convergence conjecture for polynomials in Wigner matrices

Let $p$ be a not necessarily selfadjoint polynomial in $m$ non-commuting variables. Let $X_N^{(1)},\dots,X_N^{(m)}$ be independent selfadjoint Gaussian or Wigner random matrices, converging in $\star$-moments to a free semicircular family $s_1,\dots,s_m$. Define

$$A_N:=p(X_N^{(1)},\dots,X_N^{(m)}),\qquad x:=p(s_1,\dots,s_m).$$

The convergence in $\star$-moments of the matrices to the semicircular family implies convergence in $\star$-moments of $A_N$ to $x$.

Brown measure convergence conjecture. The eigenvalue distributions $\mu_{A_N}$ converge to the Brown measure $\mu_x$ of $x$.

This conjecture seeks to establish convergence of eigenvalue distributions for polynomial functions of independent Gaussian or Wigner matrices, beyond the settings where Brown-measure continuity is already known. The source presents it as unresolved and says that the authors will address it in future work.