Conjectural square-root error bound for hyperbolic lattice counting along a geodesic

Let $l$ be a geodesic, let $N(X,l)$ be the associated hyperbolic lattice counting function, and let

$$M(X,l):=\sum_{1/2<s_j\leq 1}\frac{2}{\pi}\gamma_1(s_j)|\hat{u}_j|^2X^{s_j}$$

be its main term. Define the error term by

$$E(X,l):=N(X,l)-M(X,l).$$

Square-root error conjecture. For every $\epsilon>0$,

$$E(X,l)=O\big(X^{1/2+\epsilon}\big),$$

where the estimate depends on $l$ and $\epsilon$. The conjecture is the pointwise analogue of the proved mean-square bound, and would give essentially square-root cancellation in the hyperbolic lattice counting problem along $l$; its status is open.