Igusa's conjecture on uniform exponential-sum bounds

Let $f\in\operatorname{O}_K[x_1,\ldots,x_n]$ be a nonconstant polynomial, let $Z$ be a subscheme of $\mathbb A^n_{\operatorname{O}_K}$, and let $L$ be a non-Archimedean local field over $\operatorname{O}_K$. Write

$$E_f(Z,\psi)=\int_{\operatorname{O}_L^n}\Phi_{Z,L}(x)\psi(f(x))|dx|,$$

where $\Phi_{Z,L}$ is the characteristic function of the points reducing into $Z(k_L)$, and let $m_\psi$ denote the conductor of a nontrivial additive character $\psi$ of $L$. Igusa's conjecture. Let $\sigma>0$. If, for every such $L$ with sufficiently large residue-characteristic, there is a constant $c_L$ satisfying

$$|E_f(Z,\psi)|\leq c_L|k_L|^{-m_\psi\sigma}$$

for all additive characters of conductor $m_\psi>1$, then the constant in the corresponding uniform inequality can be chosen independently of $L$ when $k_L$ has sufficiently large characteristic. This conjecture predicts uniformity across non-Archimedean local fields and is part of the link between exponential sums, Igusa zeta functions, and arithmetic principles; no resolution is supplied in the excerpt.