Root-interlacing conjecture for eigenpolynomials of exactly solvable operators

Let $\mathfrak d(z)$ be an exactly solvable operator, and let $\{S_n(z)\}_{n\in\mathbb N,\,n\geq n_0}$ be its eigenpolynomials with $\deg S_n(z)=n$. Let $\mathfrak S_\mu$ denote the support of the corresponding asymptotic root-counting measure. Root-interlacing conjecture. The family $\{S_n(z)\}_{n\in\mathbb N,\,n\geq n_0}$ has the interlacing property along the support of its asymptotic root-counting measure: the zeros of consecutive polynomials $S_{n+1}(z)$ and $S_n(z)$ interlace along every curve segment in $\mathfrak S_\mu$ for all sufficiently high degrees $n$. The claim is motivated by numerical experiments, particularly for exactly solvable operators, and the supplied text gives no resolution.