The pentagon relation for Baxter operators in the punctured-torus skein algebra

Let $T^2-D$ be the once-punctured torus, and let $\widehat{\operatorname{Sk}}(T^2-D)$ be its completed skein algebra. For $\mathbf{x},\mathbf{y}\in\mathbb{Z}^2$, let $\omega(\mathbf{x},\mathbf{y})$ denote their intersection pairing, and let $Q_{\mathbf{x}}(v)$ and $Q_{\mathbf{y}}(w)$ be the corresponding Baxter operators. Pentagon relation. If

$$\omega(\mathbf{x},\mathbf{y})=1,$$

then

$$Q_{\mathbf{x}}(v)\cdot Q_{\mathbf{y}}(w)=Q_{\mathbf{y}}(w)\cdot Q_{\mathbf{x}+\mathbf{y}}(vw)\cdot Q_{\mathbf{x}}(v)$$

in $\widehat{\operatorname{Sk}}(T^2-D)$. This is the punctured-torus analogue of the Baxter pentagon relation established for the closed torus in the preceding theorem; the source provides the relation as a conjectural statement, and its resolution is not indicated here.