Bárány–Larman colored Tverberg conjecture

Let $F_1,\ldots,F_{d+1}$ be families of $r$ points each in $\mathds{R}^d$, considered as color classes. A colorful partition is a partition into $r$ sets $A_1,\ldots,A_r$ such that each part has exactly one point of each color. Bárány–Larman's colored Tverberg conjecture. There is a colorful partition such that

$$\bigcap_{j=1}^r \operatorname{conv}(A_j)\neq\emptyset.$$

This is a long-standing open problem concerning a colorful analogue of Tverberg's theorem.