Transfer-operator correspondence for automorphic cusp forms with unitary representations

Let $(V,\chi)$ be any unitary finite-dimensional representation of $\Gamma_\lambda$, and let $s$ satisfy $\operatorname{Re}s\in(0,1)$. Let $(\Gamma_\lambda,\chi)$-automorphic cusp forms with spectral parameter $s$ and the $1$-eigenfunctions of the transfer operator $\mathcal L_{s,\chi}$ be taken with the regularity specified in Theorem~. Transfer-operator correspondence. There is a bijection between the $(\Gamma_\lambda,\chi)$-automorphic cusp forms with spectral parameter $s$ and the $1$-eigenfunctions of $\mathcal L_{s,\chi}$. The preceding discussion explains that this would extend the known period-function characterization to arbitrary unitary finite-dimensional representations; the supplied source does not state whether the claim has been proved.