Wan's generic Newton polygon convergence conjecture

For every integer $\ell\geq 1$ and a polynomial $f(x)$, define

$$S_\ell(f)=\sum_{x\in\mathbb{F}_{p^\ell}}\zeta_p^{{\rm Tr}_{\mathbb{F}_{p^\ell}/\mathbb{F}_p}(f(x))}.$$

Let $L(f\bmod p;T)$ be the associated $L$-function, and let ${\rm NP}(f\bmod p)$ denote its $p$-adic Newton polygon. Let ${\rm HP}(f)$ be the Hodge polygon, the convex hull of the points

$$\left(k,\frac{1}{d}+\frac{2}{d}+\ldots+\frac{k}{d}\right),\qquad 0\leq k\leq d-1.$$

Wan's conjecture. There is a Zariski dense subset ${\mathcal{U}}$ in $\mathbb{A}^d$ such that, for every $f(x)\in {\mathcal{U}}$, the limit exists and

$$\lim_{p\rightarrow\infty}{\rm NP}(f\bmod p)={\rm HP}(f).$$

The Newton polygon is always above or equal to the Hodge polygon by Bombieri's theorem. The conjecture asserts convergence to the Hodge polygon for a Zariski-dense family of polynomials as the prime tends to infinity.