Asymptotic zero distribution conjecture for exceptional orthogonal polynomials

Let an exceptional orthogonal polynomial family have weight

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

where $W_0(x)$ is a classical orthogonal-polynomial weight and $\eta(x)$ is a polynomial whose degree equals the number of gaps in the exceptional degree sequence and which does not vanish on the orthogonality domain. Call the zeros lying in the orthogonality domain regular zeros, and call the remaining zeros exceptional zeros. For sufficiently high degree $n$, there are $\deg\eta$ exceptional zeros. Asymptotic zero distribution conjecture. The regular zeros of exceptional orthogonal polynomials have the same asymptotic behaviour as the zeros of their classical counterpart, and the exceptional zeros converge to the zeros of $\eta(x)$. Equivalently, after suitable normalization, the normalized counting measure of the regular zeros converges to the equilibrium measure associated with the classical counterpart. The conjecture synthesizes observed asymptotic behaviour for exceptional orthogonal polynomials. The source notes that the regular-zero claim has been proved in certain exceptional Laguerre and Jacobi cases, while the full assertion, including convergence of exceptional zeros, remains open in general.