Exotic edge-operator conjecture for analytic beta ensembles

Let $\beta>0$, let $k$ be a nonnegative integer describing an edge where the equilibrium density decays like $x^{2k+1/2}$, and let $b'_t$ denote white noise. Define the random operator

$$\mathcal{S}_{\beta,k}=-\partial_t^2+t^{\frac{1}{2k+1}}+\frac{2}{\sqrt{\beta}}t^{-\frac{k}{2k+1}}b'_t.$$

Exotic edge-operator conjecture. After the appropriate scaling, $T_n$ converges to the random operator $\mathcal{S}_{\beta,k}$. The source presents this as a conjecture and refers to a more precise formulation elsewhere; no resolution is supplied here.