Forrester–Frankel hard-edge gap-ratio asymptotic conjecture

Let $E^{\rm hard}_{\beta}(n;(0,s);\beta a/2)$ denote the hard-edge probability of finding exactly $n$ eigenvalues in $(0,s)$ for the $\beta$-ensemble, with parameter $a$, and let $\tau_{\beta a/2,\beta}^{\rm hard}(n)$ be the stated constant. For $\beta n\in\mathbb Z_{\ge 0}$, define

$$\tau_{\beta a/2,\beta}^{\rm hard}(n)={2^{-(a+n)\beta n}\over n!}\left({\beta\over2}\right)^{n(a+n-1)\beta/2}\prod_{j=1}^{\beta n}{\Gamma(a+2j/\beta)\over(2\pi)^{1/2}}\,{\prod_{j=0}^{n-1}\Gamma(1+(j+1)\beta/2)\over\prod_{j=n}^{2n-1}\Gamma(1+(j+a)\beta/2)}.$$

Hard-edge gap-ratio asymptotic conjecture. As $s\to\infty$,

$${E^{\rm hard}_{\beta}(n;(0,s);\beta a/2)\over E^{\rm hard}_{\beta}(0;(0,s);\beta a/2)}=\tau_{\beta a/2,\beta}^{\rm hard}(n)\exp\left(-\beta\left\{-\sqrt{s}\,n+\left({n^2\over2}+{na\over2}\right)\log s^{1/2}\right\}\right)\left(1+{\rm O}\left({1\over s^{1/2}}\right)\right).$$

This conjecture gives the large-$s$ asymptotics of the hard-edge spacing probabilities relative to the gap probability. The source presents the constant explicitly when $\beta n\in\mathbb Z_{\ge0}$; its validity beyond that case is not resolved here.