Maximal matrix-coefficient dimension allowing strong convergence

Determine the sharp growth regime of the matrix-coefficient dimension D_N for which strong convergence holds for noncommutative polynomials with M_{D_N}(C)-valued coefficients applied to N×N GUE or Haar unitary ensembles; specifically, close the gap between the current positive range D_N = e^{o(N)} and known counterexamples when D_N ≥ e^{C N^2}.

Background

Strong convergence with matrix coefficients is known to hold for subexponential dimensions D_N = e^{o(N)}, but can fail at dimensions e^{C N^2}. The optimal threshold between these regimes is currently unknown.

This problem connects to questions in operator space theory and to quantitative refinements of strong convergence techniques (e.g., polynomial and interpolation methods).