The Laguerre polynomial divisibility conjecture

Let $n,k\in\mathbb{N}$ and $m\in\mathbb{N}_0$. Laguerre polynomial divisibility conjecture. If the Laguerre polynomial $L_k$ divides the generalized Laguerre polynomial $L_n^{(m)}$, then

$$L_n^{(m)}=L_k.$$

The paper uses this conjecture to establish the preceding Wigner zero-set claim and reports that it is proved in several cases, while other cases remain open.