The explicit formula conjecture for Farey transfer-operator eigenfunctions

Let $q$ be a parameter and let $\mathbf{E}_{q}(y)$ denote the function introduced through the Farey transfer operators. For the previously considered cases $q=1$ and $q=\frac12$, one has $\mathbf{E}_{1}(y)=\left(\frac{2}{\pi}\right)^{\frac12}y$ and $\mathbf{E}_{1/2}(y)=y^{\frac12}$. Explicit formula conjecture. For all $q$,

$$\mathbf{E}_{q}(y)=\left(\frac{2}{\pi}\right)^{q-\frac12}y^q.$$

This extends the two explicitly verified cases and predicts a simple closed form for the functions arising from the Farey transfer operators. The supplied text gives no resolution of the assertion.