The correlation-to-symmetric-rank conjecture for multilinear forms

Let $G$ be the underlying vector space over $\mathbb{F}_2$, let $k$ be a positive integer, and write $[m]=\{1,\ldots,m\}$. For $I\subseteq[2k-2]$, let $x_I$ denote the tuple of variables indexed by $I$, and let $\operatorname{prank}$ denote partition rank. Suppose that $\alpha\colon G^{2k}\to\mathbb{F}_2$ is a multilinear form symmetric in its first $2k-1$ variables, and that

$$\alpha(x_{[2k]})+\alpha(x_{[2k-2]},x_{2k},x_{2k-1})=\sum_{I\in\binom{[2k-2]}{k-1}}\sigma(x_I,x_{2k-1})\sigma(x_{[2k-2]\setminus I},x_{2k})$$

for a symmetric multilinear form $\sigma\colon G^k\to\mathbb{F}_2$. Suppose also that $f\colon G\to\mathbb{D}$ satisfies

$$\left|\mathbb{E}_{x,a_1,\ldots,a_{2k}}\Delta_{a_1}\cdots\Delta_{a_{2k}}f(x)\,\omega^{\alpha(a_1,\ldots,a_{2k})}\right|\geq c.$$

Correlation-to-symmetric-rank conjecture. Then

$$\operatorname{prank}\sigma\leq \exp^{(O_k(1))}(O_k(c^{-1})).$$

This proposed bound is intended to resolve the paper's stated problem and, together with the arguments in the paper, yield a quantitative inverse theorem for the Gowers uniformity norms in low characteristic. The parser supplies no evidence that the problem has been resolved, so its status remains open.