Generic asymptotic normality conjecture for q-Stirling coefficients

Let $S[n,k]$, $S_B[n,k]$, $c[n,k]$, and $c_B[n,k]$ denote the four families of type $A$ and type $B$ $q$-Stirling polynomials, and view each polynomial's coefficients as a probability distribution after normalization. Generic asymptotic normality conjecture. Suppose $n,k\to\infty$ with $k/n\to\alpha$ for some $0<\alpha<1$. Then the coefficient distributions of all four families are asymptotically normal. The source proves asymptotic normality in certain fixed-$k$ regimes for the second-kind polynomials, while this proportional-growth assertion remains conjectural.