Basis conjecture for the principal right ideal of the tridiagonal-pair algebra

Let $T$ be the tridiagonal-pair algebra, let $e_0^*$ be its distinguished starred generator, and let $ZZ$ words be the zigzag words in the standard generators. Principal-ideal basis conjecture. The $\mathbb{F}$-vector space $Te_0^*$ has a basis consisting of the nontrivial $ZZ$ words that end in $e_0^*$ and do not involve $e_0$ or $e_d^*$. This conjecture gives a word basis for a principal right ideal of $T$ and is part of the paper's proposed future research program; it is not resolved in the paper.