The conjectural bound for twisted incomplete Salié sums

Let $a$ and $q$ be integers with $q\ge2$, $(a,q)=1$, and $q$ not a perfect square. Let $H,K\ge1$ and let $\lambda,\mu$ be real numbers. Write $e(t)=e^{2\pi i t}$, let $\overline h$ denote the inverse of $h$ modulo $q$, and let $\left(\frac{h}{q}\right)$ denote the relevant quadratic character. Twisted Salié-sum conjecture. For every $\epsilon>0$,

$$\sum_{\substack{1\le h\le H\\(h,q)=1}}e(\lambda h)\sum_{0\le k<K}e(\mu k)\left(\frac{h}{q}\right)e\left(\frac{a\overline h k^2}{q}\right) \ll_\epsilon \left(H^{1/2}K^{1/2}+H^{3/4}+K+q^{-1/2}HK+q^{-1/2}K^2\right)q^\epsilon.$$

This estimate is introduced as a conjectural input for the paper's treatment of twisted incomplete Salié sums; its general validity is not established in the supplied text.