Bárány's topological Tverberg conjecture

Let $d\geq 1$ and $r\geq 2$ be integers, let $N=(d+1)(r-1)$, and let $f:\Delta_N\to\mathbb{R}^d$ be continuous. A collection of faces is pairwise disjoint if the faces have no common vertices. Bárány's topological Tverberg conjecture. There exist $r$ pairwise disjoint faces $\sigma_1,\ldots,\sigma_r$ of $\Delta_N$ such that

$$f(\sigma_1)\cap\cdots\cap f(\sigma_r)\neq\emptyset.$$

The conjecture generalizes Tverberg's affine intersection theorem from affine to continuous maps. The paper explains that it is known when $r$ is a prime power, while counterexamples exist in high dimensions when $r$ is not a prime power.