Constant-coefficient recurrence conjecture for arbitrary-parameter multi-indexed polynomials

Let $P_{\mathcal{D},n}(\eta;\boldsymbol{\lambda})$ be the multi-indexed Meixner–Pollaczek or continuous Hahn polynomial, with parameters $\boldsymbol{\lambda}$ and index set $\mathcal{D}$ arbitrary. For a nonzero polynomial $Y(\eta)$, define $X(\eta)=I[\Xi_{\mathcal{D}}Y](\eta)$ and $L=\ell_{\mathcal{D}}+\deg Y+1$. Constant-coefficient recurrence conjecture. Theorem on constant-coefficient recurrence relations holds for $P_{\mathcal{D},n}(\eta;\boldsymbol{\lambda})$ with any $\boldsymbol{\lambda}$ and $\mathcal{D}$, namely these polynomials may not be orthogonal. The theorem is proved in the orthogonal setting, while the arbitrary-parameter and non-orthogonal extension is supported only by explicit calculations for small values.