Coloured Tverberg's theorem

Let $F_1, F_2, \ldots, F_{d+1}$ be colour classes, each consisting of $k$ points in $\mathbb{R}^d$. A colourful $k$-partition is a partition of their union into $k$ pairwise disjoint sets $A_1, A_2, \ldots, A_k$, each containing one point from every colour class. Coloured Tverberg's theorem. There is a colourful $k$-partition $A_1, A_2, \ldots, A_k$ such that their convex hulls intersect. This was conjectured by Bárány, Fürédi and Lovász, and in its full strength was first proposed by Bárány and Larman; it is known when $d=1,2$ or $k+1$ is prime, but the general case remains open.