Square-root upper-bound conjecture for Weyl remainder fluctuations

Let $X=\operatorname{SL}(2,\mathbb{Z})\backslash\mathcal{H}$ be the modular surface, and let $S(t)$ denote the Weyl remainder for its Maass spectrum. Square-root upper-bound conjecture. The asymptotic growth rate of the remainder fluctuations is bounded from above by

$$S(t)=o(\sqrt t).$$

The conjecture proposes an improvement over the analytically known bound $S(t)=O(t/\log t)$ and is supported in the source by numerical evidence showing that $|S(t)|/\sqrt t$ slightly decreases with $t$.