Langerman's chessboard mass partition conjecture

Let $n,d$ be positive integers. Given a finite family of $nd$ absolutely continuous measures in $\mathds{R}^d$, a set of $n$ affine hyperplanes induces a chessboard coloring: each measure is divided according to the parity of the number of chosen positive half-spaces containing a point.

Langerman's conjecture. For every such family of measures, there exists a set of $n$ hyperplanes whose induced chessboard coloring splits every measure into two equal parts.

This conjecture removes directional restrictions imposed in earlier results on chessboard colorings by hyperplane arrangements. The source notes that the case $d=n=2$ was proved by Barba, Pilz, and Schnider, while the general statement remains unresolved.