Igusa's conjecture on uniform exponential-sum bounds for homogeneous polynomials

Let $f$ be a homogeneous polynomial over $\mathbf{Q}$ in $n$ variables. For an embedded resolution $\pi$ of the relevant hypersurfaces, let $\alpha(\pi)\leq 0$ be the associated rational number, and define

$$E_f(N):=\frac{1}{N^n}\sum_{x\in\{0,\ldots,N-1\}^n}\exp\left(2\pi i\frac{f(x)}{N}\right).$$

Igusa's conjecture. There exists a constant $c$ such that for every prime $p$ and every positive integer $m$,

$$|E_f(p^m)|<c\,m^{n-1}p^{m\alpha(\pi)}.$$

This is the homogeneous-polynomial form of Igusa's conjecture: the constant in the resolution-based bound should be uniform in the prime. The supplied status evidence does not establish whether this formulation has been resolved, so it is recorded as open.