Borsuk-Ulam type theorem for the orthogonal group and orthogonal four-partitions

Given a finite Borel measure $μ$ in $\mathbb{R}^d$, when can one find $d$ mutually orthogonal hyperplanes such that \emph{every pair} of them cuts $μ$ into four equal parts? Makeev [2] stated this result and outlined a proof strategy, but the key steps were left incomplete. We give the first complet