Holmsen–Kynčl–Valculescu colored partition conjecture

Let $d\geq 2$, $\ell\geq 2$, $m\geq 2$ and $n\geq 1$ be integers with $m\geq d$ and $\ell\geq d$. Consider a set $X\subseteq\mathbb{R}^d$ of $\ell n$ points in general position, meaning that no $d+1$ points of $X$ lie in an affine hyperplane, and color its points with at least $m$ different colors. A partition of $X$ is a collection of $n$ subsets, each of size $\ell$.

Holmsen–Kynčl–Valculescu conjecture. If $X$ has a partition into $n$ subsets of size $\ell$ such that each subset contains points of at least $d$ colors, then $X$ has such a partition for which the convex hulls of the $n$ subsets are pairwise disjoint.

This conjecture concerns disjoint convex-hull partitions of colored point sets in general position and motivates the study of convex partitions capturing positive amounts from prescribed numbers of measures. The supplied source gives no resolution status.