Strichartz's almost-periodic spectral asymptotics conjecture for nonpositive curvature

Let $S$ be a surface of constant zero or negative curvature, with finite area and piecewise smooth boundary, and let $A(t)$ denote the ordinary average error in the refined spectral counting asymptotic. A function is uniformly almost periodic if it is almost periodic uniformly on the real line.

Strichartz's conjecture. There exists a uniformly almost periodic function $g$ such that

$$A(t)=g(t^{1/2})t^{-1/4}+O(t^{-1/2})$$

as $t\to\infty$. If the curvature is zero, $g$ has mean value zero. The conjecture refines the expected bounded decay of the averaged error and is supported here by numerical evidence; its extension to positive curvature is stated separately, and the source notes that the proposed expansion could not be tested in this paper.