Classification conjecture for irreducible generalized Bochner–Krall operators

Let $k$ be a positive integer, let $\partial$ denote differentiation, and let $a_j,a_m\in\mathbb C$. For a divisor $\ell$ of $k$, define

$$G:=\left(\sum_{m=0}^{\ell-1}a_m(x\partial)^m\right)\partial,$$

with $a_{\ell-1}\neq0$, and let $q(t)$ be a complex polynomial of degree $k/\ell$ with no constant term. The so-called $(k-1)$-orthogonal polynomials are generated by operators of the first displayed type. Generalized Bochner–Krall classification conjecture. For any positive integer $k$, the irreducible differential operators of order $k$ solving the generalized Bochner–Krall problem belong to one of two types: (1)

$$L=\sum_{j=1}^{k}a_jx^{j-1}\partial^j+x\partial,$$

where $a_j\in\mathbb C$ and $a_k\neq0$, generating the so-called $(k-1)$-orthogonal polynomials; or (2), with $G$ and $q$ as above,

$$L=q^{\prime}(G)G+x\partial,$$

which is an irreducible operator of order $k$. The source identifies the first type with known $(k-1)$-orthogonal polynomial constructions and attributes them to BChD; the exhaustiveness of the two-type list is the conjectural content.