Tverberg's conjecture on common flats and convex hull partitions

Let $0\leq k\leq d-1$ and let $S_0,S_1,\ldots,S_k$ be finite sets of points in $\mathbb R^d$, with

$$|S_i|=(r_i-1)(d-k+1)+1$$

for $i=0,1,\ldots,k$. A common convex-hull transversal is a $k$-flat meeting each of the relevant convex hulls. Tverberg's conjecture. Each $S_i$ can be split into $r_i$ sets $S_{i1},S_{i2},\ldots,S_{ir_i}$ such that there is a $k$-flat $F$ meeting every convex hull

$$\operatorname{conv} S_{ij},\qquad 0\leq i\leq k,\quad 1\leq j\leq r_i.$$

The conjecture generalizes Tverberg's partition theorem and is intended to place several coincidence theorems, including Rado's and the Ham-sandwich theorem, in a common framework. The paper's abstract states that it verifies the continuous version in a new case; the status of the discrete conjecture itself is not resolved by the supplied text.