Concentration conjecture for sampling the sphere by mutually orthogonal subspaces

Let $A\subseteq S^{n-1}$ be a measurable subset with $\sigma_{n-1}(A)\geq e^{-c\sqrt{n}}$. For a subspace $H\subseteq\mathbb{R}^n$ of dimension $\lfloor n/2\rfloor$, let $H^{\perp}$ be its orthogonal complement, and let $\sigma_H$ and $\sigma_{H^{\perp}}$ denote the Haar probability measures on $S^{n-1}\cap H$ and $S^{n-1}\cap H^{\perp}$, respectively. Let $\mathbb{P}_H$ denote the orthogonally invariant Haar probability measure on the Grassmannian of such subspaces. Concentration conjecture.

$$\mathbb{P}_{H}\left(\sqrt{\sigma_H(A\cap H)\sigma_{H^{\perp}}(A\cap H^{\perp})}\geq 0.9\sigma_{n-1}(A)\right) \geq 1-Ce^{-c'\sqrt{n}},$$

where $C,c,c'>0$ are universal constants. This conjecture would extend the Klartag–Regev concentration inequality to sets of measure of order $e^{-\sqrt n}$ and is intended to yield an $\Omega(\sqrt n)$ communication lower bound for the classical vector-in-subspace problem. Its resolution status is not established in the supplied text.