Friedlander–Iwaniec twisted incomplete Kloosterman-sum conjecture

Let $a,q\geq2$ be integers with $(a,q)=1$ and $q$ not a perfect square, and let $H,K\geq1$ be reals. Define $e(u):=e^{2\pi i u}$, let $\left(\frac{\cdot}{\cdot}\right)$ denote the Jacobi symbol, and let $\bar h$ denote the multiplicative inverse of $h$ modulo $q$. Friedlander–Iwaniec twisted incomplete Kloosterman-sum conjecture. For any $\epsilon>0$,

$$\begin{aligned} &\mathop{\sum_{1\leq h\leq H}}_{(h,q)=1}\sum_{0\leq k<K}\left(\frac{h}{q}\right)e\left(\frac{a\bar h k^2}{q}\right)\\ &\ll_\epsilon\left(H^{1/2}K^{1/2}+H^{3/4}+K+q^{-1/2}HK+q^{-1/2}K^2\right)q^\epsilon. \end{aligned}$$

The source invokes this estimate as an assumption behind stronger bounds for quadratic fractional parts; its resolution status is not supplied.