Conjecture on joint spectral measures for rank-two graph pairs

Let $\Gamma$ be a finite subgroup of $GL(2,\mathbb{Z})$, let $(\mathcal{G}^{\Gamma}_{\rho_1},\mathcal{G}^{\Gamma}_{\rho_2})$ be the associated pair of graphs, let $\mathbb{T}^2$ be the parameter torus, and let $\mathfrak{D}$ be the corresponding domain of $\Gamma$-invariant variables $(x,y)$. Write $S_{\varrho}(\theta)$ for the orbit function and $J_{\Gamma}(x,y)$ for the Jacobian associated with the change to these invariant variables. Joint spectral-measure conjecture. The joint spectral measure $\varepsilon$ over $\mathbb{T}^2$ for the pair of graphs is

$$\mathrm{d}\varepsilon(\omega_1,\omega_2) = \frac{1}{|\Gamma|} \, |S_{\varrho}(\theta)|^2 \, \mathrm{d}\omega_1 \, \mathrm{d}\omega_2.$$

The joint spectral measure $\widetilde{\nu}$ over $\mathfrak{D}$ for the same pair is

$$\mathrm{d}\widetilde{\nu}(x,y) = \frac{|S_{\varrho}(\theta)|^2}{|J_{\Gamma}(x,y)|} \, \mathrm{d}x \, \mathrm{d}y.$$

These formulas propose a uniform description of the joint spectral measures associated with the rank-two Lie-group and finite-subgroup graph constructions. The supplied text gives no evidence that the conjecture has been proved or refuted.