Polynomial $U^3(\mathbb F_2^n)$-inverse conjecture

Let $\mathbb F_2^n$ be the $n$-dimensional vector space over $\mathbb F_2$, let $f\in L^\infty(\mathbb F_2^n)$, and let $\|\cdot\|_{U^3(\mathbb F_2^n)}$ and $\|\cdot\|_{L^\infty(\mathbb F_2^n)}$ denote the $U^3$ and supremum norms. A quadratic polynomial is a map $q:\mathbb F_2^n\to\mathbb F_2$ of degree at most two.

Polynomial $U^3(\mathbb F_2^n)$-inverse conjecture. If

$$\|f\|_{U^3(\mathbb F_2^n)}\geq\delta\|f\|_{L^\infty(\mathbb F_2^n)},$$

then there is a quadratic polynomial $q:\mathbb F_2^n\to\mathbb F_2$ such that

$$|\langle f,(-1)^q\rangle_{L^2(\mathbb F_2^n)}|\geq\exp(-O(\log\delta^{-1}))\|f\|_{L^\infty(\mathbb F_2^n)}.$$

This is the polynomial-strength inverse statement associated in the source with Marton's conjecture for $\mathbb F_2^n$; the excerpt describes the currently available bound as $\exp(-O(\log^{3+o(1)}\delta^{-1}))$.