The $L^2$ anti-concentration conjecture for symmetric log-concave measures

Let $d\in\mathbb{N}$. Let $H_n$ be the hyperoctahedral group acting on $\mathbb{R}^n$ by coordinate permutations and sign changes. A measure $\mu$ on $\mathbb{R}^n$ is isotropic if it has mean zero and identity covariance matrix, and $H_n$-invariant if it is invariant under this group. For a degree-$d$ homogeneous polynomial $f$, write $\int f^2\,d\mu$ for its squared $L^2(\mu)$-norm. The conjecture asserts that there is a constant $C_d>0$, depending only on $d$, such that for every isotropic, log-concave, $H_n$-invariant measure $\mu$ on $\mathbb{R}^n$ and every degree-$d$ homogeneous polynomial $f:\mathbb{R}^n\to\mathbb{R}$,

$$\int_{\mathbb{R}^n} f^2\,d\mu \geq C_d.$$

Consequently, for odd $d$, since homogeneous functions of odd degree are odd, one has $\operatorname{Var}_\mu(f)\geq C_d$. This would extend the known degree-three result and clarify the absence of bad examples for higher-degree symmetric measures, while the general case remains open.