Character-sum conjecture for the extended-support problem

Let $P\geq 1$, let $k\leq P$ be a nonsquare, so that the character $\left(\frac{k}{\cdot}\right)$ is nonprincipal, and set $H=P^{2/3+\delta}$. Let $e(x)=\exp(2\pi i x)$ and let $\overline p$ denote the inverse of $p$ in the modulus $k^2$. Character-sum conjecture. There exist $\delta>0$ and $\varepsilon>0$ such that

$$\sum_{h\leq H}\sum_{p\leq P}\left(\frac{k}{p}\right)e\left(\frac{h^3\overline p}{k^2}\right)\ll P^{1-3\delta/2-\varepsilon}.$$

Such cancellation would support extending the one-level density beyond the currently available Fourier-transform support and is proposed as evidence for orthogonal symmetry in the family of all elliptic curves; its resolution is not supplied here.