Polynomial inverse Gowers conjecture for the $U^3$ norm

Let $f:\mathbb{F}_2^n\to\mathbb{F}_2$ be a Boolean function, and let $\|f\|_{U^3}$ denote its Gowers $U^3$ norm. Polynomial inverse Gowers conjecture for the $U^3$ norm. If

$$\|f\|_{U^3}\ge\epsilon,$$

then there exists a quadratic polynomial $p:\mathbb{F}_2^n\to\mathbb{F}_2$ such that

$$\Pr[f(x)=p(x)]\ge\tfrac{1}{2}+\epsilon^{O(1)}.$$

The source notes that the inverse Gowers statement for $U^3$ was known with a non-polynomial quantitative bound, specifically of exponential type, and conjectures a polynomial dependence on $\epsilon$. The claim is open in the source context.