Conjecture on singular support and L^1-convergence of the absolutely continuous part for c < 0

Prove that, for the Brownian motion on the unitary quantum group U_N^+ at times t_c = N ln(√2 N) + c N with c < 0, the singular part of the large-N limit lies outside the support of the semicircular distribution ν_SC, and that the absolutely continuous part converges to that of the free Meixner law η_c in L^1(ν_SC).

Background

After establishing the cutoff profile for c ≥ 0 and a lower bound for c < 0, the authors formulate a conjecture to complete the description in the loss-of-absolute-continuity regime. They propose that the singular part resides outside the support of ν_SC and that the absolutely continuous part converges in L^1(ν_SC) to that of η_c even when c < 0.

This conjecture, if proved, would settle the profile for c < 0 by confirming true L^1-convergence of the absolutely continuous component and clarifying the relationship between the singular support and ν_SC.