The at-most-two non-real zeros conjecture for normalised polynomial sequences

Let $\{W_n(z)\}_n$ be a normalised polynomial sequence satisfying the recurrence

$$W_n(z)=(az+b)W_{n-1}(z)+(cz+d)W_{n-2}(z),$$

with $a,b,d<0$ and $c>0$. A normalised polynomial sequence is one whose recurrence and initial polynomials satisfy the normalisation specified in the paper.

At-most-two non-real zeros conjecture. Every polynomial $W_n(z)$ has at most two non-real zeros.

This conjecture is proposed as a generalisation of the paper's real-rootedness results. It is partially supported by the theorem asserting real-rootedness of every $W_n(z)$ when $c$ is sufficiently large, but its validity for all $c>0$ remains open.