Hard-edge universality conjecture for products of coupled non-Gaussian matrices

Let $\alpha>0$ and let $m$ be a fixed nonnegative integer. Let $\delta_1,\ldots,\delta_m$ and $\delta$ be complex numbers with absolute value less than $\alpha$, and let $X_1$ and $X_2$ be the two complex random matrices described by conditions (C1)--(C4): their vector pairs are independent with the stated identical-distribution structure, have the stated zero means and second moments, have the variances and pairwise covariances prescribed by (C3), and possess finite fourth moments. For

$$\alpha=\frac{1+\mu}{2\mu},\qquad \delta=\frac{1-\mu}{2\mu},\qquad 0<\mu\leq 1,$$

Non-Gaussian hard-edge universality conjecture. Under assumptions (C1)--(C4), the conclusion of Theorem (hardlimits) should still hold, with $\kappa=0$. This conjecture extends the hard-edge limiting behavior beyond Gaussian entries; the source notes that the Gaussian joint density considered earlier satisfies these assumptions, while the corresponding universality result for general entries remains open.