Square-root cancellation conjecture for spectral exponential sums on arithmetic hyperbolic surfaces

Let $X>2$, let $\Gamma$ be an arithmetic Fuchsian group defining a hyperbolic surface, and let $\mathcal{S}(T,X)$ denote the spectral exponential sum associated with that surface. Square-root cancellation conjecture. Uniformly in $X$, up to a factor of order $X^{\epsilon}$, one has

$$\mathcal{S}(T,X) \ll T^{1+\epsilon}X^{\epsilon}.$$

This conjecture generalizes the Petridis–Risager conjecture for $PSL_{2}(\mathbb{Z})\backslash\mathbb{H}$ and predicts square-root cancellation in the spectral parameter. The supplied context gives no resolution, so the conjecture remains open.