Existence of a finite-term recurrence relation for eigenpolynomials

Let a differential operator be given as in $$, and let ${P_n}$ be its sequence of eigenpolynomials. The coefficients ${\alpha_{n,k}}$ are those occurring in the recurrence relation denoted by $$. Existence conjecture. There exist an integer $p\leq N-1$ and a sequence of coefficients ${\alpha_{n,k}}$ such that $$ holds. The claim predicts a finite-term recurrence relation for the eigenpolynomials; the supplied text gives no resolution status or further context establishing whether it is proved or remains open.