Ramos's Grünbaum–Hadwiger–Ramos mass partition conjecture

Let $d,k,m$ be positive integers. A triple $(d,k,m)$ is a solution of the Grünbaum–Hadwiger–Ramos mass partition problem if, for any $m$ absolutely continuous finite measures in $\mathds{R}^d$, there exist $k$ affine hyperplanes dividing $\mathds{R}^d$ into $2^k$ parts of equal size in each measure.

Ramos's conjecture. The triple $(d,k,m)$ is a solution if and only if

$$d \ge \left\lceil \left( \frac{2^k-1}{k}\right)m\right\rceil.$$

Ramos's dimension condition is necessary by an argument using measures concentrated near the moment curve. The conjecture asks whether this necessary condition is also sufficient; the survey gives no resolution of the general problem.