Bounded Gotsman–Linial conjecture for semi-algebraic sets

Let $S\subseteq\mathbb{R}^d$ be a semi-algebraic set of description complexity $r$, and let $a_1,\dots,a_n\in\mathbb{R}^d$. Define $f:\\{-1,1\\}^n\to\\{0,1\\}$ by letting $f(\xi_1,\dots,\xi_n)$ indicate whether

$$a_1\xi_1+\dots+a_n\xi_n\in S.$$

Let $\operatorname{AS}(f)$ denote the average sensitivity of $f$. Bounded Gotsman–Linial conjecture. There is a constant $C_{r,d}$ depending only on $r$ and $d$ such that

$$\operatorname{AS}(f)\le C_{r,d}\sqrt n.$$

A full resolution of the Gotsman–Linial conjecture would imply the main forward conjecture for affine varieties, but the paper notes that this bounded semi-algebraic version would already suffice.