The covering conjecture for localized skein algebras

Let $\Sigma$ be an unpunctured surface, let $\Delta$ be an ideal triangulation of $\Sigma$, and for each internal edge $E\in e_{\mathrm{int}}(\Delta)$ let $\mathscr{S}^q_{\mathfrak{sl}_3,\Sigma}[(\Delta;E)^{-1}]$ be the Ore localization obtained by inverting the elementary webs along the edges of $\Delta$ other than $E$, together with $A^{1/2}$. Covering conjecture. For every ideal triangulation $\Delta$ of $\Sigma$,

$$\mathscr{S}^q_{\mathfrak{sl}_3,\Sigma}[\partial^{-1}] = \bigcap_{E\in e_{\mathrm{int}}(\Delta)}\mathscr{S}^q_{\mathfrak{sl}_3,\Sigma}[(\Delta;E)^{-1}].$$

This conjecture would identify the boundary-localized skein algebra as the intersection of the codimension-one localizations and is used in the paper to derive its equality with the upper cluster algebra.