Conjecture on Airy limits for solvable random block tridiagonal matrix ensembles

Let $r\geq 2$ and let $\beta=1$ or $2$. Consider the point process associated with the $r=2$, $\beta s=2$ joint density in Theorem 1, and the point process associated with the $r=2$, $\beta s=4$ joint density when specified below. Airy scaling-limit conjecture. More generally, the point process scaling limit of the first density is distributed as $\operatorname{Airy}_{\beta+2/r}$ for all $r\geq 2$ and $\beta=1$ or $2$. For the second density with $\beta s=4$ and $\beta=1$, the point process scaling limit is $\operatorname{Airy}_3$. These claims extend the explicitly established classical cases, in which the limiting processes are $\operatorname{Airy}_2$ and $\operatorname{Airy}_4$, and conjecturally identify the diffusion characterization with the general $\operatorname{Airy}_{\beta\gamma}$ process.