B. Shapiro's zero-location conjecture for generalized polynomial recurrences

Let $A(z)$ and $B(z)$ be arbitrary complex polynomials, and let $k$ and $\ell$ be coprime integers with $1\leq \ell<k$. Let $\\{P_n(z)\\}_{n=0}^{\infty}$ satisfy

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

for $n=1,2,\dots$, with $P_0(z)=1$ and $P_{-1}(z)=\dots=P_{1-k}(z)=0$. Let $\mathcal C$ be the real algebraic curve defined by

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

Shapiro's conjecture. Every zero of every $P_n(z)$ that is not a zero of $A(z)$ or $B(z)$ lies on $\mathcal C$. This generalizes Tran's zero-location conjecture to recurrences with lag $\ell$. The case $\ell=1$ was settled, but the conjecture for coprime $1<\ell<k$ remains open and is supported by numerical experiments.