Bárány–Larman colorful Tverberg conjecture

Let $r,d$ be positive integers. Given $d+1$ sets $X_1, \ldots, X_{d+1}$, each consisting of $r$ points of $\mathds{R}^d$, consider partitions of their union into $r$ sets $A_1, \ldots, A_r$. Bárány–Larman colorful Tverberg conjecture. There exists such a partition in which each $A_j$ has exactly one point of each $X_i$ and the convex hulls of $A_1, \ldots, A_r$ intersect. The conjecture is the colorful version of Tverberg's theorem. It has been confirmed when $r+1$ is prime or when $d=2$, while the general case is not resolved in the supplied text.