Polynomial bias-rank conjecture

Let $k$ be a finite field of characteristic $p$, let $V$ be a finite-dimensional $k$-vector space, and let $P:V\to k$ be a polynomial of degree $d$ and rank greater than $r$. The bias-rank relation asserts that the rank threshold giving a prescribed bias bound can be polynomial in the inverse bias exponent. Bias-rank conjecture. For $p>d$, the rank threshold satisfies $r=s^{-O_d(1)}$. This conjecture is known for $d=2,3$, while the general case remains open.