The conjectural bound for the spectral exponential sum over the Picard manifold

Let $\Gamma=\mathrm{PSL}_{2}(\mathcal{O})$, let $\lambda_j=1+t_j^2$ be the eigenvalues of the hyperbolic Laplacian on $\Gamma\backslash\mathfrak{h}^{3}$, and let $S(T,X)$ be the spectral exponential sum defined in the paper. Spectral exponential-sum conjecture. For every $\epsilon>0$ and $X\gg1$, one has

$$S(T,X)\ll_{\epsilon}T^{2+\epsilon}X^{\epsilon}.$$

The conjecture predicts cancellation of order $T$ in the three-dimensional spectral sum and is intended to imply an error term of essentially order $X^{1+\epsilon}$ for the prime geodesic theorem. The paper presents numerical evidence but does not establish the bound.