Brown measure convergence under strong convergence (conjectural)

Prove that if a family of random matrices X^N strongly converges to a limiting family x in a C*-probability space, then for every noncommutative polynomial P the empirical distribution of the complex eigenvalues of P(X^N,(X^N)*) converges to the Brown measure of P(x,x*).

Background

The Brown measure provides the natural notion of complex eigenvalue distribution in operator-algebraic settings. It is conjectured that strong convergence should force convergence of empirical complex spectra to Brown measures for non-normal polynomials.

This would parallel the self-adjoint case, where weak and strong convergence control spectral distributions and edges, but at present this conjectural principle is only known in special cases.