Weak Gotsman–Linial conjecture on polynomial threshold function influence

Let $f:\{-1,1\}^n\to\{-1,1\}$ be a degree-$d$ polynomial threshold function, and let $\operatorname{\mathbf{I}}[f]$ denote its total influence. Weak Gotsman–Linial conjecture. For any degree-$d$ threshold function $f$ on $n$ variables,

$$\operatorname{\mathbf{I}}[f]\le O(d\sqrt{n}).$$

This is a relaxation of the refuted exact Gotsman–Linial maximizer conjecture: for $d<\sqrt n$, the conjectured symmetric bound has order $\Theta(d\sqrt n)$, and the weaker asymptotic upper bound remains the relevant question in the paper.