Conjecture on Bessel limits for solvable random block tridiagonal ensembles

Let $\operatorname{Bessel}_{\beta,a}$ be the hard-edge point process defined by the $r=1$, $\gamma=1$ limiting operator, and consider the solvable ensembles with the explicit joint densities in equations (density3) and (density4). In the known cases, the smallest points converge to $\operatorname{Bessel}_{2,a/2}$ when $r=2$, $\beta s=2$, $\beta=1$, and to $\operatorname{Bessel}_{4,a/2}$ when $r=2$, $\beta s=4$, $\beta=2$. Bessel scaling-limit conjecture. For density3, the minimal points have scaling limit

$$\operatorname{Bessel}_{\beta+2/r,\,a/(1+2/(\beta r))}$$

for $r\geq 2$ and $\beta=1$ or $2$. For density4 with $\beta s=4$ and $\beta=1$, the scaling limit is instead $\operatorname{Bessel}_{3,a/3}$. This conjecture extends the explicitly identified classical hard-edge limits to the corresponding nonclassical Bessel processes.