Classification of unavoidable Tverberg partitions

Let $T(d,r)$ be the Tverberg number, and let a Tverberg type be a partition $\mathcal I=\{I_1,\ldots,I_r\}$ of $[T(d,r)]$ into $r$ parts. Let $\operatorname{Tv}_{\mathcal I}(P)$ be the $T(d,r)$-ary predicate stating that the Tverberg type $\mathcal I$ is a Tverberg partition for $P$. The type $\mathcal I$ is colorful if, for each $1\leq i\leq d+1$, the $r$ consecutive integers $\{(r-1)(i-1)+1,\ldots,(r-1)i+1\}$ belong one to each of the parts $I_1,\ldots,I_r$.

Classification conjecture. For every Tverberg type $\mathcal I$, if $\mathcal I$ is colorful, then $\operatorname{Tv}_{\mathcal I}$ is unavoidable; otherwise, $\neg\operatorname{Tv}_{\mathcal I}$ is unavoidable.

The conjecture classifies Tverberg-type predicates according to whether they are unavoidable or avoidable. The supplied text also contains a theorem asserting exactly this classification, so the candidate appears to be a restatement or transcription issue rather than an unresolved conjecture; the source's resolution status should be checked.