Stochastic Bessel operator hard-edge singular-value conjecture

Let $\mathcal{J}_{a}^{\beta}$ denote the stochastic Bessel operator in its original form, and let $\tilde{\mathcal{J}}_{a}^{\beta}$ denote its Liouville normal form. Consider type (i) and type (ii) boundary conditions, and let $k$ be a positive integer. Stochastic Bessel hard-edge conjecture. Under type (i) boundary conditions, the $k$th least singular value of the stochastic Bessel operator follows the $k$th hard edge distribution with parameters $\beta$ and $a$. Under type (ii) boundary conditions, the hard edge distribution has parameters $\beta$ and $a+1$. This is true both for the original form, $\mathcal{J}_{a}^{\beta}$, and for Liouville normal form, $\tilde{\mathcal{J}}_{a}^{\beta}$. The conjecture is motivated by the finite-difference schemes for these operators and the convergence of the corresponding small singular values to hard-edge distributions.