Short sums conjecture for Kloosterman sums

Let $p$ be prime, let $a\in\mathbb{F}_p^\times$, let $N$ and $H$ be integers, and write $\overline{x}$ for the multiplicative inverse of $x$ modulo $p$. Short sums conjecture. There exists an $\varepsilon>0$ such that

$$\left|\sum_{N\leq x\leq N+H}e\left(\frac{ax+\overline{x}}{p}\right)\right|\ll H^{1-\varepsilon},$$

uniformly for any $1<N<p$, $p^{1/2-\varepsilon/2}<H<p^{1/2+\varepsilon/2}$, and $a\in\mathbb{F}_p^\times$. This is a conjectural power-saving bound for incomplete Kloosterman sums in the transition range $H\asymp p^{1/2}$; the parser supplies no evidence that it has been resolved.