Ismail's structure conjecture for orthogonal polynomials and the Askey–Wilson operator

Let $(P_n)_{n\geq 0}$ be a monic orthogonal polynomial sequence (OPS), and let $\pi$ be a polynomial independent of $n$. For the first assertion, assume that $\pi$ has degree at most $2$ and that

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

Ismail's structure conjecture. Then $(P_n)_{n\geq 0}$ are continuous $q$-Jacobi polynomials, Al-Salam–Chihara polynomials, or special or limiting cases of these families. The same conclusion should hold when $\pi$ has degree $s+1$ and, for positive integers $r$ and $s$,

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

with $\pi$ independent of $n$. The first part is attributed to M. E. H. Ismail; the second part is disproved by a counterexample to the displayed general relation.