Resolve the geometric-mean conjecture for VSP

Prove or refute the conjecture that, for every sufficiently large measurable subset A of the sphere, the geometric mean of its measures on a Haar-random half-dimensional subspace and its orthogonal complement is at least 0.9 times its global spherical measure except with probability exponentially small in the square root of the dimension.

Background

A geometric-mean concentration conjecture proposed by Grupel would imply Ω(√n) lower bounds for two-way VSP protocols in a substantial regime. The paper explains that this conjecture avoids the spherical-cap obstruction affecting an earlier rectangle-bound approach, but only partial cases are known.