Gowers's true complexity conjecture for systems of linear forms

Let $L_1,\dots,L_m$ be a system of linear forms in $d$ variables over $\mathbb{F}_p^n$. Its true complexity is the least integer $k$ such that, for every $\eta>0$, there exists $\epsilon>0$ with the property that whenever $A\subset\mathbb{F}_p^n$ has density $\delta$ and $\\|A-\delta\mathbf{1}\\|_{U^{k+1}}<\epsilon$, the normalized count of the tuples $L_1(x_1,\dots,x_d),\dots,L_m(x_1,\dots,x_d)$ in $A$ is within $\eta$ of the random-set expectation, assuming there are no degeneracies. Gowers's true complexity conjecture. The true complexity of the system is the least integer $k$ such that the forms $L_1^{k+1},L_2^{k+1},\dots,L_m^{k+1}$ are linearly independent. This conjecture identifies the required uniformity degree with the first degree at which the powers of the forms become linearly independent; the source later records that the conjecture is false in this generality.