Strong inverse conjecture for the Gowers norm

Let $p$ be prime, $k\geq1$, $G=\mathbb{F}_p^n$, and $\mathbb D=\{z\in\mathbb C:|z|\leq1\}$. Let $\varepsilon:\mathbb N\to\mathbb R^+$ be decreasing, and let $\mathrm{Poly}^k(G)$ be the degree-at-most-$k$ polynomial phases on $G$. Strong inverse conjecture for the Gowers norm. For every $\eta>0$, there exists $M=M(k,p,\eta,\varepsilon)$ such that, if $f:G\to\mathbb D$ satisfies $\|f\|_{U^{k+1}(G)}\geq\eta$, then for a uniformly random tuple $\vec h=(h_1,\dots,h_M)\in G^M$, with probability at least $1/2$ there exist $1\leq m\leq M$, $P\in\mathrm{Poly}^k(G)$, and a function $F:{\mathbb D}^{\mathbb F_p^M}\to\mathbb C$ of Lipschitz constant at most $M$ such that

$$\left|\mathbb E_{x\in G}f(x)e(-P(x))\right|\geq\frac1m$$

and

$$\left|\mathbb E_{x\in G}e(P(x))-F\left(\left(f(x+a\cdot\vec h)\right)_{a\in\mathbb F_p^M}\right)\right|\leq\varepsilon(m).$$

This is presented as a strengthening of the inverse conjecture, motivated by the claim that the correlating polynomial is approximately measurable from the function and random shifts. The source explains that the Bergelson–Tao–Ziegler conjecture would imply this stronger statement, but does not state that the strong version has been proved.