Ismail's characterization conjecture for orthogonal polynomial sequences on the Askey–Wilson lattice

Let $(P_n)_{n\geq 0}$ be a monic orthogonal polynomial sequence and let $\pi$ be a polynomial of degree at most $2$ independent of $n$. Let $\mathcal{D}_q$ denote the Askey–Wilson operator on the lattice $x=(q^{-s}+q^s)/2$. Suppose that

$$\pi(x)\mathcal{D}_qP_n(x)=(a_nx+b_n)P_n(x)+c_nP_{n-1}(x).$$

Ismail's characterization conjecture. Then $(P_n)_{n\geq 0}$ are continuous $q$-Jacobi polynomials, Al-Salam–Chihara polynomials, or special or limiting cases of these families. More generally, if $\pi$ has degree $s+1$ and the preceding condition is replaced by

$$\pi(x)\mathcal{D}_qP_n(x)=\sum_{k=-r}^{s}c_{n,k}P_{n+k}(x),$$

for positive integers $r$ and $s$ and a polynomial $\pi$ independent of $n$, the same conclusion holds. This is a proposed characterization of the orthogonal polynomial sequences satisfying first-order relations involving the Askey–Wilson operator; the source presents it as Ismail's conjecture, and no resolution is supplied here.