Counterexamples to the Ramos conjecture for two hyperplanes

For every $n\ge 2$, we construct $4n-2$ nondegenerate Gaussian measures on ${\mathbb{R}}^{6n-3}$ that cannot be simultaneously equipartitioned by two affine hyperplanes. This disproves the Ramos conjecture for two hyperplanes. Combined with known upper bounds, the construction shows that $3\cdot 2^{s-1}-2$ is the least dimension guaranteeing a common two-hyperplane equipartition of $2^s-2$ absolutely continuous probability measures, for every $s\ge 3$. We characterize the Gaussian equipartition threshold in terms of the least number of positive definite quadratic measurements needed for phase retrieval. Modified complex polynomial multiplication gives $2r-2$ positive definite measurements in every even dimension $r\ge 4$. This number is optimal when $r=2^k+2$, $k\ge 1$.