The topological Bárány-Larman conjecture for prime numbers

The Bárány-Larman conjecture states that for any $d+1$ sets of $r$ points each in ${\mathbb{R}}^d$, considered as color classes, we can partition their union into $r$ rainbow $(d+1)$-tuples whose convex hulls intersect. We prove that the topological version of this conjecture holds when $r$ is a prime number. We also show that the optimal colorful Tverberg theorem of Blagojević, Matschke, and Ziegler cannot be extended to prime powers.