The inverse theorem for the $U^{s+1}$ Gowers norm on finite abelian groups

Let $G$ be a finite additive group, let $ta > 0$, let $s eq 1$, and let $f \colon G \to \mathbb{C}$ be a $1$-bounded function with $\|f\|_{U^{s+1}(G)} \geq \eta$. A degree $s$ filtered nilmanifold is a nilmanifold equipped with a filtration of degree $s$; a polynomial map is understood with respect to this filtration. The inverse theorem for $U^{s+1}(G)$. There exists a degree $s$ filtered nilmanifold $H/\Gamma$, drawn from some finite collection ${\mathcal N}_{s,\eta}$ of such nilmanifolds depending only on $s,\eta$ and not on $G$, a Lipschitz function $F \colon H/\Gamma \to \mathbb{C}$ of Lipschitz norm $O_{\eta,s}(1)$, and a polynomial map $g \colon G \to H/\Gamma$ such that

$$|\mathbb{E}_{x \in G} f(x) \overline{F(g(x))}| \gg_{\eta,s} 1.$$

This conjecture predicts a uniform nilsequence inverse theorem for arbitrary finite additive groups, with the target nilmanifold drawn from a finite family independent of $G$.