Kuijlaars–Milson conjecture on the asymptotics of exceptional polynomial zeros

Let $\{p_n(x)\}_{n=m}^{\infty}$ be a family of exceptional orthogonal polynomials with weight function

$$W(x)=\frac{W_0(x)}{\eta(x)^2},$$

where $W_0(x)$ is the weight function of a classical orthogonal polynomial family and $\eta(x)$ is a polynomial that does not vanish on the domain of orthogonality. The zeros of the exceptional polynomials consist of regular zeros in the domain of orthogonality and exceptional zeros outside it; the degree of $\eta$ equals the number of exceptional zeros. Kuijlaars–Milson conjecture. The regular zeros of XOPS have the same asymptotic behaviour as the zeros of their classical counterpart, while the exceptional zeros converge to the zeros of the denominator polynomial $\eta(x)$. This conjecture concerns the large-degree zero distribution of exceptional orthogonal polynomials; the source presents the paper's results for the $X_1$-Jacobi and $X_1$-Laguerre families as quantitative contributions toward it, but does not state that the general conjecture is resolved.