Interlacing conjecture for polynomial recurrences on the equimodular discriminant

Let $p_{n+1}(z)=\sum_{i=1}^{k}Q_i(z)p_{n-i}(z)$ be a finite recurrence with fixed polynomial coefficients, and suppose $\deg p_n(z)=n$ for $n\in\mathbb N$. Let $\Xi_Q$ be the equimodular discriminant of the associated asymptotic symbol equation. Interlacing conjecture. For all sufficiently large $n$, the zeros of $p_{n+1}(z)$ and $p_n(z)$ interlace along $\Xi_Q$. More precisely, after choosing a sufficiently small neighborhood $N(\Xi_Q)$ and an identification with the normal bundle to $\Xi_Q$, there is $n_0$ such that the projected roots of $p_n$ and $p_{n+1}$ interlace for every $n\geq n_0$. The claim is motivated by computer experiments and is formulated for complex zeros along the discriminant curve; the supplied text gives no resolution evidence.