Neitzke–Yan compatibility conjecture for quantum UV–IR maps

Let $\Gamma=H_1(\widetilde{\Sigma};\mathbb{Z})$ have its standard intersection pairing, and let

$$2\Gamma=\{2\gamma\mid\gamma\in\Gamma\},\qquad \Gamma^{\mathrm{odd}}=\{\gamma\in\Gamma\mid\sigma(\gamma)=-\gamma\},\qquad \Gamma^{\mathrm{even}}=\{\gamma\in\Gamma\mid\sigma(\gamma)=\gamma\},$$

where $\sigma$ is the $\mathbb{Z}/2$-deck transformation action. Let $Q_{2\Gamma}$, $Q_{\Gamma^{\mathrm{odd}}}$, and $Q_{\Gamma^{\mathrm{even}}}$ be the corresponding quantum tori, and let $F_{\tau}$, $\pi$, $\rho$, $F^{\mathrm{odd}}$, and $F^{\mathrm{even}}$ be the maps specified in the diagram and formulas below. Neitzke–Yan compatibility conjecture. The diagram

\begin{tikzcd} \operatorname{SkAlg}^{\mathfrak{gl}_2,*}_{q}(\Sigma) \arrow[d, "\pi"] \arrow[r, "F_{\tau}"] & \operatorname{SkAlg}^{\mathfrak{gl}_1,*}_{q}(\widetilde{\Sigma}_{\tau}) \cong Q_{2\Gamma} \arrow[d, "\rho"]\\ \operatorname{SkAlg}^{\mathfrak{sl}_2}_{A}(\Sigma)\otimes\operatorname{SkAlg}^{\mathfrak{gl}_1,*}_{-A}(\Sigma) \arrow[r, "F^{\mathrm{odd}}\otimes F^{\mathrm{even}}"] & Q_{\Gamma^{\mathrm{odd}}}\otimes Q_{\Gamma^{\mathrm{even}}} \end{tikzcd}

is commutative, where

$$\rho(x_{2\gamma})=x_{\gamma-\sigma(\gamma)}\otimes x_{\gamma+\sigma(\gamma)},$$ $$F^{\mathrm{odd}}([L]^{\mathfrak{sl}_2})=(-A)^{-w(\vec L)}\rho^{\mathrm{odd}}\bigl(\iota\circ F([\vec L]^{\mathfrak{gl}_2})\bigr),$$

with $\rho^{\mathrm{odd}}(x_{2\gamma})=x_{\gamma-\sigma(\gamma)}$, and

$$F^{\mathrm{even}}([\vec L]^{\mathfrak{gl}_1})=(-A)^{w(\vec L)}x_{p^{-1}(\vec L)}.$$

Moreover, $F^{\mathrm{odd}}$ coincides with the Bonahon–Wong quantum trace map.