The Mathieu-subspace conjecture for generalized Laguerre polynomials

Let ${\bf k}\in\mathbb N^n$, and let $L^{[\bf k]}_\alpha(z)$ denote the generalized Laguerre polynomials for multi-indices $\alpha\in\mathbb N^n$. Let $\mathcal M$ be the subspace of the polynomial algebra $\mathcal A[z]$ spanned by the polynomials $L^{[\bf k]}_\alpha(z)$ with $0\ne\alpha\in\mathbb N^n$. A subspace of a commutative algebra is a Mathieu subspace if, whenever all positive powers of an element lie in it, multiplication by any fixed algebra element preserves membership for all sufficiently large powers. The generalized Laguerre Mathieu-subspace conjecture. For every ${\bf k}\in\mathbb N^n$, $\mathcal M$ is a Mathieu subspace of $\mathcal A[z]$. This is identified as a special case of a broader conjecture for classical orthogonal polynomials. The source notes that it remains open even for the classical Laguerre polynomials in one variable, corresponding to ${\bf k}=0$.