Hard edge statistics conjecture for products involving Pólya ensembles

Let $G$ be a complex square matrix satisfying conditions (1) and (2), and let $H$ be a random Hermitian matrix drawn from a polynomial ensemble with limiting Green function $G(x)$ such that $0<|G(0)|<\infty$. Write $K_\infty^{(G)}$ for the limiting hard edge kernel associated with $G$, and let $\Theta$ denote the Heaviside function, $\operatorname{Re}$ and $\operatorname{Im}$ the real and imaginary parts, and $\operatorname{sign}$ the sign function. Then the limiting hard edge kernel of the eigenvalues of $GHG^*$ should be

Hard edge statistics conjecture.

$$\begin{aligned} K_\infty(\tilde{a}_1,\tilde{a}_2)={}&\Theta(-\operatorname{Re}[G(0)]\tilde{a}_2)|\operatorname{Re}[G(0)]|K_\infty^{(G)}(\operatorname{sign}(\tilde{a}_2)|\operatorname{Re}[G(0)]|\tilde{a}_1,|\operatorname{Re}[G(0)]\tilde{a}_2|)\\ &+\operatorname{Im}[G(0)]\int_{-1}^{1}\frac{dt}{2\pi}J\omega^{(\infty)}\left(\tilde{a}_1(\operatorname{Im}[G(0)]t-i\operatorname{Re}[G(0)])\right)K\omega^{(\infty)}\left(\tilde{a}_2(\operatorname{Im}[G(0)]t-i\operatorname{Re}[G(0)])\right). \end{aligned}$$

The conjecture proposes that the hard edge limit obtained for the GUE extends, with the Green function evaluated at zero replacing the GUE-specific quantity, to products involving a broad class of polynomial ensembles. The precise assumptions on $G$, the referenced conditions, and the limiting functions are inherited from the paper; the parser supplies no evidence that this conjecture has been resolved.