Inverse conjecture for the Gowers norm

Let $p$ be a prime and let $k\geq 1$. For a finite abelian group $G$, define the Gowers uniformity norm by

$$\|f\|_{U^{k+1}(G)}^{2^{k+1}}\coloneqq \mathbb{E}_{x,h_1,\dots,h_{k+1}\in G}\prod_{\omega\in\{0,1\}^{k+1}}\mathcal{C}^{|\omega|}f(x+\omega\cdot\vec h).$$

Here $\mathcal{C}$ denotes complex conjugation, $\omega\cdot\vec h=\omega_1h_1+\cdots+\omega_{k+1}h_{k+1}$, and $e(\theta)=e^{2\pi i\theta}$. For $G=\mathbb{F}_p^n$, let $\mathrm{Poly}^k(G)$ denote the group of non-classical polynomials $P:G\to\mathbb{T}$ of degree at most $k$, and let ${\mathbb D}=\{z\in\mathbb{C}:|z|\leq 1\}$. Inverse conjecture for the Gowers norm. For every $\eta>0$ there exists $c=c(k,p,\eta)>0$ such that, whenever $f:G\to{\mathbb D}$ satisfies $\|f\|_{U^{k+1}(G)}\geq\eta$, there exists $P\in\mathrm{Poly}^k(G)$ such that

$$\left|\mathbb{E}_{x\in G}f(x)e(-P(x))\right|\geq c.$$

This conjecture is stated in the source as established for all $k$ and $p$, with earlier proofs in several ranges and subsequent alternate proofs; it is therefore a theorem rather than an open conjecture.