The zigzag-word basis conjecture for the algebra $\tilde T$

Fix an integer $d\geq 0$ and a sequence $p\in\operatorname{Feas}(d,\mathbb F)$. Let $\tilde T=\tilde T(p,\mathbb F)$ be the algebra defined in the paper, and let zigzag words mean the distinguished words in its generators. Zigzag-word basis conjecture. The $\mathbb F$-vector space $\tilde T$ has a basis consisting of its zigzag words. This is the formulation given for the directness conjecture in the setting of $\tilde T$; the supplied text gives no resolution status.