Comparison conjecture for slow and fast transfer operators of Hecke triangle groups

Let $\Gamma$ be a cofinite Hecke triangle group, let $s$ satisfy $\operatorname{Re} s=\tfrac12$, and let $\mathcal L_{H,s}^{+}$ and $\mathcal L_{H,s}^{-}$ be the meromorphic extensions of the even and odd transfer operators, respectively. Let $\operatorname{FE}_s(\mathbb R^{+})^{\mathrm{dec},+}_{\omega}$ and $\operatorname{FE}_s(\mathbb R^{-})^{\mathrm{dec},-}_{\omega}$ denote the corresponding spaces of period functions.

Comparison conjecture. The space of $1$-eigenfunctions of the meromorphic extension of $\mathcal L_{H,s}^{+}$ is linearly isomorphic to $\operatorname{FE}_s(\mathbb R^{+})^{\mathrm{dec},+}_{\omega}$ and hence corresponds to even Maass cusp forms. The space of $1$-eigenfunctions of the meromorphic extension of $\mathcal L_{H,s}^{-}$ is linearly isomorphic to $\operatorname{FE}_s(\mathbb R^{-})^{\mathrm{dec},-}_{\omega}$ and hence corresponds to odd Maass cusp forms.

This is proposed as a comparison between the slow and fast discrete dynamical systems arising from the geometric construction. The supplied text does not state whether the comparison has subsequently been proved.