Soberón–Takahashi conjecture on bisection by parallel hyperplanes

Let $d$ and $k$ be positive integers. A finite absolutely continuous measure on $\mathbb{R}^d$ is given for each of $d+k-1$ measures.

Soberón–Takahashi conjecture. Any $d+k-1$ finite absolutely continuous measures on $\mathbb{R}^d$ can be simultaneously bisected by the chessboard coloring induced by $k$ or fewer parallel hyperplanes.

This conjecture concerns simultaneous mass bisection by parallel hyperplanes and generalizes the ham sandwich theorem. The case $k=2$ has been proved by Soberón and Takahashi and later by Blagojević and Crabb; the general conjecture remains open.