Normal-distribution conjecture for additive twists on Fuchsian groups

Let $\Gamma$ be a Fuchsian group of the first kind with cusps, and let $\phi$ be a Maaß cusp form for $\Gamma$ of integer weight. For $\Gamma=\operatorname{SL}(2,\mathbb{Z})$, the relevant multisets of normalized additive-twist central values become distributed, as the denominator bound tends to infinity, to a centered normal law. Normal-distribution conjecture. The statement of the additive-twists-on-average theorem for $\operatorname{SL}(2,\mathbb{Z})$ remains true when $\Gamma$ is replaced by an arbitrary Fuchsian group of the first kind with cusps and $\phi$ is a Maaß cusp form for $\Gamma$ of integer weight. The paper proves the averaged result for $\operatorname{SL}(2,\mathbb{Z})$ and conjectures that the restriction to that group is artificial.