The superweak Gotsman–Linial conjecture

Let $f:\{-1,1\}^n\to\{-1,1\}$ be an $(n,d)$-PTF, meaning the sign of a real polynomial of degree at most $d$ on the Boolean hypercube. Let $\mathbf{AS}[f]$ denote its average sensitivity.

Superweak Gotsman–Linial conjecture. For some function $g$ depending only on $d$,

$$\mathbf{AS}[f]\in O\left(\sqrt n\log^{g(d)}n\right).$$

Daniel Kane resolved this conjecture. It is weaker than the $O(d\sqrt n)$ asymptotic bound and was introduced as a still useful consequence for applications.