Extension of the main theorem to degrees at least the characteristic

Let $p$ be a prime and let $d$ be a positive integer with $d\geq 2$. The main theorem concerns homogeneous polynomials and multilinear forms over $\mathbb{F}_p$ under the assumption $d<p$.

Extension conjecture. Theorem still holds for $d\geq p$.

The authors note that the restriction $d<p$ is used to ensure $d!\neq 0$ and hence to associate a unique multilinear form to a homogeneous polynomial. They expect this assumption can be removed, as it was removed in related work of Green and Tao by Kaufman and Lovett; the extension remains open here.