Stable-limit conjecture for finite free Appell polynomials

Let $f$ be in the Laguerre--Pólya class and satisfy the regular-variation condition on its roots. Let

$$A_d(x)=f(D)x^d,$$

and set $\rho=\frac{1}{\alpha}-1$. Write $\mu_p$ for the empirical root distribution associated with a polynomial $p$, and let $\mathcal{D}_c$ denote the dilation operator by $c$. Stable-limit conjecture. The measures

$$\mu_{\mathcal{D}_{h(d)^{-1}d^\rho}A_d}$$

converge weakly as $d\to\infty$ to a $\boxplus$-stable distribution with stability parameter $\alpha$. This conjecture proposes an analytic finite-free-probability analogue of stable-limit results, connecting Laguerre--Pólya functions, Appell polynomials, and free infinite divisibility; its status is not resolved in the supplied source.