Fouvry's conjectural short exponential-sum estimate

Let $k\geqslant0$ be an integer, let $a,h,m$ be integers with $h\neq0$, $m\geqslant1$, and $2\nmid am$, and let $N$ be real with $m\leqslant N\leqslant m^2$. Let $N_1$ satisfy $N<N_1\leqslant2N$, and write $\overline n$ for the multiplicative inverse of $n$ modulo $m^2$ whenever $(m,n)=1$. For $e(z)=\exp(2\pi i z)$, consider the sum over integers $n$ with $N<n\leqslant N_1$ and $n\equiv a\pmod{4^k}$.

Fouvry's conjectural estimate. There exists an absolute constant $\vartheta\in[\frac12,1)$ such that, uniformly under these conditions,

$$\sum_{\substack{N<n\leqslant N_1\\ n\equiv a\pmod{4^k},\ (m,n)=1}}e\left(\frac{h\overline n^2}{m^2}\right)\ll_k(h,m^2)^{\frac12}N^{\vartheta}.$$

This estimate is introduced as a conjectural input for controlling short algebraic exponential sums in the study of Pell equations. The source gives no resolution of the conjecture; its strength lies in asserting a power-saving exponent $\vartheta<1$ uniformly across the stated ranges.