Gowers' linear forms and uniformity conjecture

Let $L_1,\dots,L_r$ be a system of linear forms in $x=(x_1,\dots,x_s)$, and let $G$ be either $\mathbb{Z}_N$ or $\mathbb{F}_p^n$ for sufficiently large $p$. Suppose that the $k$th powers of the forms $L_i$ are linearly independent. For bounded functions $f,g$ on $G$, consider the multilinear averages

$$\mathbb{E}_x\prod_{i=1}^r f(L_i(x))$$

and

$$\mathbb{E}_x\prod_{i=1}^r g(L_i(x)).$$

Gowers' conjecture. The two averages are close whenever $\\|f-g\\|_{U^k}$ is small.

This conjecture would identify the minimal uniformity norm for which the multilinear expression associated with the system of forms is uniformly continuous, namely the smallest $k$ for which the $k$th powers of the forms are linearly independent. The converse is not hard to prove; the conjectured forward implication remains the substantive open direction.