Soberón's colorful Tverberg tolerance conjecture

Let $r,d$ be fixed positive integers, let $N$ be a positive integer, and let $p_r$ be the probability that a random permutation of $r$ numbers has at least one fixed point. For $N$ color classes of $r$ points each in $\mathds{R}^d$, a colorful partition is a partition into $r$ sets with one point of each color in every set; it remains a Tverberg partition after removing color classes if the convex hulls of the remaining parts intersect. Soberón's conjecture. In the known colorful Tverberg-with-tolerance result, the factor $p_r$ is unnecessary: the optimal value of the tolerance is $t=N-o(N)$. The conjecture concerns strengthening the known bound $t=p_rN-O(\sqrt{N\ln N})$ so that almost all color classes may be removed while preserving a colorful Tverberg partition; the supplied text gives no resolution status.