The limiting 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 $f^*_{n,1}$ be the symmetric degree-one candidate defined by the monic polynomial with its root at the integer closest to $0$ of parity opposite to $n$, and let $\mathbf{AS}[f]$ denote average sensitivity.

Limiting Gotsman–Linial conjecture. The average sensitivity satisfies

$$\mathbf{AS}[f]\leq d\,\mathbf{AS}[f^*_{n,1}].$$

The paper presents this as a revised conjecture after refuting the exact Gotsman–Linial maximization claim. It would imply the remaining cases discussed there, but the supplied text gives no resolution.