Quadratic repulsion conjecture for determinants of independent Gaussian analytic functions

Let $z_0$ be a point in the bulk of the pseudospectrum, and let $\tilde G_{z_0}$ be the determinant of the $J\times J$ matrix of independent Gaussian analytic functions arising as the scaling limit of the perturbed operator. Let $d^{k,M}_{z_0}(w_1,\ldots,w_k)$ denote the $k$-point density of the zero point process of $\tilde G_{z_0}$. Quadratic repulsion conjecture. For any compact set $O\Subset\mathbb{C}$, there exists a constant $C>1$ depending only on $O$ and $k$ such that, for all pairwise distinct points $w_1,\ldots,w_k\in O$,

$$C^{-1}\prod_{i<j}|w_i-w_j|^2\leq d^{k,M}_{z_0}(w_1,\ldots,w_k)\leq C\prod_{i<j}|w_i-w_j|^2.$$

This conjecture predicts quadratic short-range repulsion for the limiting eigenvalue process generated by matrix-valued Gaussian analytic functions, in contrast with the different repulsion behavior of the corresponding scalar Gaussian analytic function process. Its status is not resolved in the source.