Forrester–Frankel asymptotic conjecture for Fisher–Hartwig Hankel determinants

Let $m,N\in\mathbb{N}$, $q\in(-1/2,\infty)^m$, and let $\mu_1,\dots,\mu_m\in\operatorname{int}(\operatorname{supp}\rho)$, where $\rho(x)$ is the limiting expected eigenvalue density described above. Suppose that either $\Omega=\mathbb{R}$ or $\Omega=(0,\infty)$, and let $\omega_N(x)=e^{-NV(x)}$, where $V(x)$ is an $N$-independent polynomial with positive leading coefficient and no zeros in $\Omega$. Define

$$\mathcal{H}_{N,m,q}(\mu):=\frac{H_{N,N,m,q}(\mu)}{H_{N+|q|,N}},\qquad |q|:=q_1+\dots+q_m.$$

Forrester–Frankel conjecture. As $N\to\infty$ with $m$, $q$, and $\mu$ fixed,

$$\mathcal{H}_{N,m,q}(\mu)=N^{\sum_{i=1}^m(q_i^2-q_i)}\prod_{i=1}^m[\omega_N(\mu_i)]^{-q_i}\prod_{i=1}^m\frac{G^2(q_i+1)}{G(2q_i+1)}(2\pi)^{q_i^2-q_i}$$ $$\times\prod_{1\le j<k\le m}|\mu_k-\mu_j|^{-2q_jq_k}\prod_{i=1}^m[\rho(\mu_i)]^{q_i^2}\,[1+o(1)].$$

Here $G$ denotes the Barnes $G$-function. This conjecture gives the predicted leading asymptotics of large Hankel determinants with Fisher–Hartwig singularities in the bulk of the limiting eigenvalue density; its status is left unresolved by the supplied source and parser evidence.