Factorization conjecture for concentrated homogeneous polynomials

Let $x_1,\dots,x_n$ be independent and uniformly chosen from $\{-1,1\}$, and let

$$f(x_1,\dots,x_n)=\sum_{1\leq i_1\dots\leq i_k}a_{i_1\dots i_k}x_{i_1}\dots x_{i_k}$$

be a degree-$k$ homogeneous polynomial with at least $mn^{k-1}$ nonzero coefficients. Polynomial factorization conjecture. If the concentration satisfies

$$\sup_c\mathbb{P}(f(x_1,\dots,x_n)=c)=\Omega_k(m^{-k/2+\epsilon}),$$

then $f$ differs in only a few coefficients from a polynomial which factors. The same passage also conjectures the upper bound $\sup_c\mathbb{P}(f(x_1,\dots,x_n)=c)=O(m^{-1/2})$ under the stated coefficient-count hypothesis.