The classification conjecture for classical Bochner–Krall polynomial sequences

Let $P_n$ be a sequence of monic polynomials obtained as eigenpolynomials of a linear differential operator solving the classical Bochner–Krall problem. Let $T$ and $T^{-1}$ denote the forward and backward shift operators, respectively, and let $u(n)$ and $v(n)$ be the coefficients in the three-term recurrence, with difference operator

$$\Lambda(n):=T+u(n)\operatorname{Id}+v(n)T^{-1}.$$

A finite Darboux transformation means a finite sequence of Darboux transformations applied to this difference operator. The classical Bochner–Krall classification conjecture. Every such sequence belongs to one of two classes: (1) a classical sequence of orthogonal polynomials with, in general, complex-valued parameters—Hermite polynomials $H_n$, Laguerre polynomials $L_n^{(\alpha)}$, Jacobi polynomials $P_n^{(\alpha,\beta)}$, or Bessel polynomials $Y_n$; or (2) a sequence obtained from Laguerre polynomials $L_n^{(\alpha)}$ with positive integer $\alpha$, or from Jacobi polynomials $P_n^{(\alpha,\beta)}$ with positive integer $\alpha$, $\beta$, or both, by applying a finite number of Darboux transformations to $\Lambda(n)$. The first family of cases is classical, while the listed transformed families are known to solve the problem; the open difficulty is proving that this list is exhaustive.