Tidor's conjecture on approximately symmetric multilinear forms

Let $G$ be a vector space over $\mathbb{F}_p$, and let $\alpha\colon G^k\tomathbb{F}_p$ be an $r$-approximately symmetric multilinear form, meaning that the partition rank of $\alpha-\alpha\circ\pi$ is at most $r$ for every permutation $pi\in\operatorname{Sym}_k$. A multilinear form is symmetric if it is invariant under all permutations of its arguments. Tidor's conjecture. There exists a symmetric multilinear form $\sigma\colon G^k\tomathbb{F}_p$ such that

$$\operatorname{prank}(\sigma-\alpha)\leq O(r^{O(1)}),$$

where the implicit constants may depend on $p$ and $k$. Tidor formulated this conjecture in the study of the inverse problem for the Gowers uniformity norm in low characteristic and proved it for trilinear maps; the general case remains open in the supplied source.