Quadratic relation conjecture for Askey–Wilson polynomials

Let $n$ be a positive integer, let $x$ be a variable, and let $a$, $b$, $c$, $d$, and $q$ be parameters. Let $p_n(x;a,b,c,d;q)$ denote the Askey–Wilson polynomial. Quadratic relation conjecture. The following quadratic relation should hold:

$$\begin{aligned} &ab(1-q^{n-1})(1-cdq^{n-2})p_n(x;a,b,c,d;q)p_{n-2}(x;aq,bq,c,d;q)\\ &=(1-abq^{n-1})(1-abcdq^{n-1})p_{n-1}(x;a,b,c,d;q)p_{n-1}(x;aq,bq,c,d;q)\\ &\quad-(1-ab)(1-abcdq^{2n-2})p_{n-1}(x;aq,b,c,d;q)p_{n-1}(x;a,bq,c,d;q). \end{aligned}$$

This conjecture generalizes the quadratic relation obtained in the special case of Askey–Wilson parameters $(c,-c)$. A direct proof would provide an alternative route to the determinant identity developed in the paper; the authors instead use a more general theorem, so the status of this proposed relation remains open in the source.