Roger–Yang skein algebra positivity conjecture for punctured surfaces

Let $\Sigma$ be a punctured surface with puncture set $\mathcal{M}$, and let $\mathsf{B}$ be the $\mathbb{Z}[q^{\pm1/2}]$-basis of the Roger–Yang skein algebra $\mathscr{S}_q(\Sigma)$ obtained from the normalized sequences $(\overline{T}_n(x))$ described in the paper. A punctured surface is non-small when it is not a sphere with at most three punctures. Roger–Yang positivity conjecture. For any non-small punctured surface $(\Sigma,\mathcal{M})$ with $|\mathcal{M}|\geq 2$, the basis $\mathsf{B}$ is a positive basis of the Roger–Yang skein algebra $\mathscr{S}_q(\Sigma)$. This conjecture asserts positivity of the distinguished basis beyond the cases established in the paper; the general claim remains open.