Local square mean in the hyperbolic circle problem and sums of Salié sums

Let $\mathrm{\Gamma }\subseteq PSL(2,\mathbb{R})$ be a finite volume Fuchsian group. The hyperbolic circle problem is the estimation of the number of elements of the $\mathrm{\Gamma }$-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 $e^{\frac{2}{3}R}$ is known, and this has not been improved for any group. Recently, taking $z=w$ and considering $\mathrm{\Gamma }=PSL(2,\mathbb{Z})$, we have shown the estimate $e^{(\frac{9}{14}+\epsilon )R}$ 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 Salié sums.