Rank conjecture for principal submodules of the tridiagonal-pair algebra

Let $R=\mathbb{F}[x_1,\ldots,x_d]$, and let $e_i,e_i^*$ be the standard generators of $T$. Principal-submodule rank conjecture. For $0\le i\le d$, the $R$-submodules $e_i^*Te_0^*$ and $e_iTe_0^*$ are both free with rank

$${d\choose i}.$$

This conjecture refines the proposed free-module description of $Te_0^*$ by predicting the ranks of its generator-indexed submodules; it is stated as an open future-research problem in the paper.