Universality of the stochastic Airy operator at nonregular spectral edges

Assume $V$ is nonregular. For $k\geq 1$, let $\psi_V(t)\sim (\mathcal{E}-t)^{\genfrac{}{}{}{}{4k+1}{2}}$ as $t\uparrow \mathcal{E}=\mathcal{E}(0)$, where $\mathcal{E}$ is the rightmost edge of the support of $\psi_V$. Define $c$ by

$$\lim_{\epsilon\downarrow 0}\genfrac{}{}{}{}{\mathcal{E}-\mathcal{E}(\epsilon)}{\epsilon^{\genfrac{}{}{}{}{1}{2k+1}}}=c,$$

and set $\gamma=c^{-2/3}(\mathcal{E}/2)^{-1/3}$. Let $T_n$ be the associated tridiagonal operator and let $W'$ denote white noise. Define

$$H_{n,k}=\gamma n^{\genfrac{}{}{}{}{2}{4k+3}}(\mathcal{E}I-T_n).$$

Universality conjecture. The operator $H_{n,k}$ converges in the sense of Theorem (i) to

$$\mathcal{S}_{\beta,k}=-\genfrac{}{}{}{}{d^2}{dx^2}+x^{\genfrac{}{}{}{}{1}{2k+1}}+\genfrac{}{}{}{}{2}{\sqrt{\beta}}x^{-\genfrac{}{}{}{}{k}{2k+1}}W'(x),$$

on the half-line with Dirichlet conditions at the origin. In particular, the ordered eigenvalues and eigenvectors of $H_{n,k}$ converge jointly in law to those of $\mathcal{S}_{\beta,k}$.

This asserts universality of the limiting stochastic differential operator at a nonregular spectral edge, extending the stochastic Airy operator framework beyond the regular case. The parser supplies no evidence that the statement has been resolved, so its status is left open.