Langerman's conjecture on bisection by hyperplanes

Let a mass distribution in $\mathbb{R}^d$ be a measure for which the bisection notion is defined. A set of $n$ hyperplanes bisects a mass distribution when, after orienting the hyperplanes, the two checkerboard regions determined by the parity of the number of positive sides have equal mass. Langerman's conjecture. Any $dn$ mass distributions in $\mathbb{R}^d$ can be simultaneously bisected by $n$ hyperplanes. This generalizes the Ham-Sandwich theorem to several hyperplane cuts and the checkerboard bisection induced by their arrangement. The conjecture has been proved for several values of $n$ and $d$, but remains open in full generality.