The colorful Tverberg partition conjecture

Fix integers $r\geq 2$ and $d\geq 1$. A partition $X_1\sqcup\cdots\sqcup X_r$ of $\{1,2,\dots,(r-1)(d+1)+1\}$ is colorful if, for every $1\leq k\leq d+1$, the block

$$Y_k=\{(r-1)(k-1)+1,\dots,(r-1)k+1\}$$

contains exactly one element of each $X_i$. A partition occurs as a Tverberg partition in a sequence $x_1,\dots,x_N$ if some subsequence of length $(r-1)(d+1)+1$ has the corresponding convex hulls with a common point. Colorful Tverberg partition conjecture. Any colorful partition occurs as a Tverberg partition in every sufficiently long sequence of generic points in $\mathbb{R}^d$, whereas every noncolorful partition fails to occur in some arbitrarily large point sets.

The conjecture formalizes the claim that asymptotically only colorful Tverberg partitions are forced. The source presents the colorful-partition behavior on the stretched moment curve as understood and records this asymptotic assertion as conjectural.