Commutativity conjecture for powers of the raising and lowering maps

Let $U_0$ be the subspace used in the paper's decomposition associated with a tridiagonal pair, and let $L$ and $R$ denote the corresponding lowering and raising maps. Commutativity conjecture. The maps

$$LR,L^2R^2,\ldots,L^dR^d$$

mutually commute on $U_0$. The claim is established in the supplied discussion for the first few powers and is stated as needed in general; no general resolution is given in the text.