Plancherel–Rotach edge asymptotic conjecture for orthonormal polynomials

Let $\{\widehat P_k(x)\}_{k=1}^n$ be a sequence of polynomials orthonormal with respect to a weight function $w(x)$ on an unbounded interval, and let $t_n$ be the large transition point of $\widehat P_n(x)$. Assume the interval is $(0,\infty)$ or $(-\infty,\infty)$. Suppose that

$$-\log w(x)=O(x^m)\qquad\text{as }x\to\infty.$$

Then the Plancherel–Rotach edge asymptotic conjecture asserts that

$$t_n=O\left(n^{1/m}\right)\qquad\text{as }n\to\infty.$$

Moreover, uniformly for bounded real $s$, with $x=t_n(1+sn^{-2/3})$,

$$w(x)^{1/2}\widehat P_n(x)\sim \hat c(s)n^k(1+o(1))\qquad\text{as }n\to\infty,$$

where $\hat c(s)$ is uniformly bounded in $s$ and

$$k=\frac16-\frac{1}{2m}.$$

The conjecture predicts the transition-point scale and the edge amplitude for a broad class of orthonormal polynomials with weights having polynomial logarithmic decay; the source provides the formulation but no resolution, so its status is open.