Twisted Linnik–Selberg conjecture for Salié sums

Let $T(m,n;c)$ denote the Salié sum for odd positive $c$, and let $\theta$ be the inverse of $L$ modulo $c$ when it exists. For every $\epsilon>0$ there is a $\delta>0$ such that, whenever $C eq 1$, $f$ is smooth and supported in $[C,2C]$, $L,K,r$ are positive integers with $2\mid L$ and $\gcd(L,Kr)=1$, $B>0$, and

$$LK(1+B)\ll_{\delta}C^{\delta},\qquad x^jf^{(j)}(x)\ll_{\delta,j}C^{j\delta}$$

for all $x>0$ and integers $j\geq 0$, then for all positive integers $m,n$ and every $\alpha\in[-B,B]$,

$$\frac{1}{C}\sum_{\substack{c\equiv r\pmod L\K\mid c}}f(c)T(m,\overline{L}n;c)e\left(\frac{\sqrt{mn}}{c}\alpha\right)\ll_{\epsilon,\delta}(mnC)^{\epsilon}.$$

This estimate is the conjectural input intended to improve the local square-mean bound in the hyperbolic circle problem; its resolution status is not specified in the supplied text.