Zigzag-word basis conjecture for the tridiagonal-pair algebra

Let $T$ be the algebra generated by the standard generators $\{e_i\}_{i=0}^d$ and $\{e_i^*\}_{i=0}^d$. A word in these generators is zigzag (or $ZZ$) when it is alternating and satisfies the two non-between conditions stated in the paper. Zigzag-word basis conjecture. For fixed integers $r,s$ with $0\le r,s\le d$, the $\mathbb{F}$-vector space $T$ has a basis consisting of the $ZZ$ words that do not involve $e_r$ or $e_s^*$. The conjecture proposes an explicit basis for $T$ in terms of zigzag words; it appears in the paper's suggestions for future research and is not resolved there.