Factorization conjecture for concentrated multilinear forms

Let $k$ be a fixed positive integer. Let $y_1,\dots,y_k$ be independent vectors uniformly chosen from $\{-1,1\}^n$, let $A$ be a $k$-multilinear form whose coefficients are all nonzero, and let $f$ be a function of $k-1$ variables. Multilinear factorization conjecture. If some $\epsilon>0$ satisfies

$$\mathbb{P}\bigl(A(y_1,\dots,y_k)=f(y_2,\dots,y_k)\bigr)\geq n^{-k/2+\epsilon},$$

then there is a partition of $\{y_1,\dots,y_k\}$ into disjoint sets $S$ and $T$, and functions $f_1$ and $f_2$, such that $f_1$ depends only on the variables in $S$, $f_2$ only on the variables in $T$, and $A$ differs from $f_1f_2$ in $o(n^2)$ coefficients. This conjectures that concentration substantially above the generic multilinear scale is explained by an approximately factorizable form.