Polynomial-rank symmetrization conjecture for multilinear forms

Let $V$ be a vector space over $\mathbb F_p$, let $k$ be a positive integer, and let $T\colon V^k\to\mathbb F_p$ be a $k$-linear form. For each permutation $\pi\in\mathfrak S_k$, write $T_\pi$ for the form obtained by permuting the arguments of $T$, and let $\operatorname{prank}$ denote partition rank. A form is symmetric if it is invariant under all permutations of its arguments.

Symmetrization conjecture. If

$$\operatorname{prank}(T-T_\pi)\leq r$$

for every $\pi\in\mathfrak S_k$, then there exists a symmetric $k$-linear form $S\colon V^k\to\mathbb F_p$ such that

$$\operatorname{prank}(S-T)\leq r',$$

where $r'$ is a polynomial function of $r$.

The conjecture concerns quantitative stability of symmetry under partition rank. It is only interesting when $p\leq k$, since for $p>k$ averaging over permutations gives a symmetric form with the required bound. Combined with work of Gowers and Milićević, it would provide the missing ingredient for quantitative bounds in the $U^{p+1}$-inverse theorem; the source presents it as surprisingly difficult and potentially independently interesting.