Transfer-operator eigenfunctions determining resonance residues

Let $\Gamma_\lambda$ be the Hecke triangle group, let $\chi$ be a unitary finite-dimensional representation, and let $\mathcal L_{s,\chi}$ be the associated transfer operator. At a resonance $s$, consider the sufficiently regular vectors in the $1$-eigenspace of $\mathcal L_{s,\chi}$. Residue-determination conjecture. These sufficiently regular $1$-eigenfunction vectors determine the residues at the resonance $s$. This expectation is motivated by the preceding theorems and is supported in the source by a later theorem; the supplied material does not establish that the conjecture itself is fully resolved.