Log-squared weight conjecture for q-scaled recurrence coefficients

Let $p_n(x)$ be symmetric orthonormal polynomials satisfying

$$xp_n(x)=a_{n+1}p_{n+1}(x)+a_np_{n-1}(x).$$

Assume that $a_nq^{an}$ has a limit as $n\to\infty$, for some $a>0$, and let $w$ be a weight function for the $p_n$.

Log-squared weight conjecture. The weight satisfies

$$\ln w(x)\approx-c\ln^2|x|,\qquad\text{for some }c>0,$$

as $x\to\infty$ or $x\to-\infty$, but not necessarily for both.

This conjecture concerns the weak confinement and indeterminate moment problems characteristic of the $q$-random matrix ensembles. The source presents it as being based on observations; no resolution is supplied here.