Improved error-term conjecture for the limit form of the trace formula

Let $X$ be the asymptotic parameter in the terms of the limit form of the trace formula, and let $f_\infty$ and $f_q$ denote the test-function components appearing there. For every $\varepsilon>0$, the error term $o(X)$ in all those terms is conjectured to admit the stronger bound $O(X^{1/2+\varepsilon})$, with an implied constant depending only on $f_\infty$, $f_q$, and $\varepsilon$.

Improved error-term conjecture. In all terms above, the error term $o(X)$ can be replaced by $O(X^{1/2+\varepsilon})$ for any $\varepsilon>0$, where the implied constant only depends on $f_\infty$, $f_q$ and $\varepsilon$.

This conjecture concerns the expected square-root-scale improvement in the error terms of the limit form of the trace formula for $\mathsf{GL}_2$ over $\mathbb{Q}$ with the specified ramification. The supplied text gives no evidence that the improvement has been proved or disproved.