Local square mean in the hyperbolic circle problem and sums of Sali\'e sums

Let $\Gamma\subseteq PSL(2, \mathbb R)$ be a finite volume Fuchsian group. The hyperbolic circle problem is the estimation of the number of elements of the $\Gamma$-orbit of $z$ in a hyperbolic circle around $w$ of radius $R$, where $z$ and $w$ are given points of the upper half plane and $R$ is a large number. An estimate with error term $e^{\frac 23R}$ is known, and this has not been improved for any group. Recently, taking $z=w$ and considering $\Gamma = PSL(2, \mathbb Z)$, we have shown the estimate $e^{\left(\frac 9{14}+\epsilon\right)R}$ for the local $L^2$-norm of the error term, which is better than the pointwise bound. Here we improve the exponent $\frac 9{14}$, conditionally on a twisted Linnik-Selberg-type conjecture for sums of Sali\'e sums.