Large-argument asymptotic conjecture for a multivariate hypergeometric function

Let ${}_0F_1^{(\beta/2)}({}_\_;a+2n;({s_0\over4})^{\beta a/2},({s_1\over4})^\beta,\dots,({s_n\over4})^\beta)$ be the generalized hypergeometric function appearing in the hard-edge scaling limit, and let the parameters $a$, $n$, and $\beta$ be as in the source. The corresponding right-hand side is the explicit expression obtained from the scaled large-$N$ asymptotic in equation (4.23a).

Large-argument hypergeometric asymptotic conjecture. For large values of the arguments $\{s_j\}_{j=0,\dots,n}$, the generalized hypergeometric function has the asymptotic form given by the right-hand side of (4.23a), up to terms that vanish as the arguments tend to infinity.

This conjecture connects the double-scaling limit of the $\beta$-ensemble average with the large-argument asymptotics of the generalized hypergeometric function. The source does not provide a resolution of the conjecture.