Eventual real-rootedness conjecture for GKP polynomial sequences

Let $u,v$ be parameters, let $P$ be the polynomial associated with the sequence, and let $q_n^{u,v}$ be the corresponding polynomial sequence. Assume

$$|v|+u<0$$

and

$$P\bigl(\pm(u+v-2l)\bigr)\ne 0\quad\text{for }l\in\mathbb{N}.$$

Eventual real-rootedness conjecture. For all sufficiently large $n$, every zero of $q_n^{u,v}$ is real.

The conjecture predicts eventual real-rootedness even when $P$ has non-real zeros. A preceding example shows that without the relevant assumption, the associated polynomials can have non-real zeros for every $n$; the source gives no resolution of this conjecture.