Cluckers–Nguyen uniform bound conjecture for exponential sums

Let $f$ be a polynomial in $n$ variables, with $s$ and $d$ as in the preceding setup, and define

$$E_f(\boldsymbol\psi)=\frac{1}{N^n}\sum_{x\in (\mathbb Z/N\mathbb Z)^n}\psi(f(x)).$$

Here $N\geq 1$ and $\psi:\mathbb Z/N\mathbb Z\to\mathbb C^\times$ is a group monomorphism. Cluckers–Nguyen's conjecture. For each $\varepsilon>0$, there is a constant $C_\varepsilon>0$ such that

$$|E_f(\psi)|\leq C_\varepsilon N^{-\frac{n-s}{d}+\varepsilon}$$

for all integers $N\geq 1$ and all such group monomorphisms $\psi$. This conjecture seeks a uniform analogue over finite rings of the Deligne–Katz bound for exponential sums and is connected with Igusa's conjecture on exponential sums and the Hasse principle; its resolution is not specified here.