Tran's conjecture on roots of polynomial recurrence sequences

Let $A(z)$ and $B(z)$ be arbitrary polynomials, and let $\{P_n(z)\}_{n=1}^{\infty}$ satisfy the recurrence of length $k$

$$P_n(z)+B(z)P_{n-1}(z)+A(z)P_{n-k}(z)=0$$

with standard initial conditions $P_0(z)=1$ and $P_{-1}(z)=\dots=P_{-k+1}(z)=0$, where $A(z)\neq 0$. Define the real semialgebraic curve $\mathcal C\subset\mathbb C$ by

$$\operatorname{Im}\left(\frac{B^k(z)}{A(z)}\right)=0$$

and

$$0\leq (-1)^k\operatorname{Re}\left(\frac{B^k(z)}{A(z)}\right)\leq \frac{k^k}{(k-1)^{k-1}}.$$

Tran's conjecture. Every zero of every polynomial $P_n(z)$ lies on $\mathcal C$, and the zeros become dense in $\mathcal C$ as $n\to\infty$.

This conjecture proposes an exceptional finite-$n$ localization phenomenon for polynomial recurrence sequences: unlike the general Beraha–Kahane–Weiss setting, their zeros should already lie on the limiting real algebraic curve. The paper presents it as a generalization of Tran's earlier conjecture; the parser supplies no evidence resolving it.