The AP conjecture on Tverberg prescribable dimensions

Let $r\geq 2$ and $d\geq 1$ be integers. An $r$-tuple $(d_1,\dots,d_r)$ is Tverberg admissible if

$$\left\lfloor\frac d2\right\rfloor\leq d_1\leq\cdots\leq d_r\leq d$$

and $\sum_i d_i=(r-1)d$. AP conjecture. Every Tverberg admissible $r$-tuple is Tverberg prescribable for $r$ a power of a prime: there is a sufficiently large simplex $\Delta$ such that every continuous map $f\colon\Delta\longrightarrow\mathbb{R}^d$ identifies $r$ points from pairwise disjoint faces $\sigma_1,\dots,\sigma_r$ with $\dim\sigma_i=d_i$.

The conjecture extends known balanced cases and is motivated by the question of which face dimensions can be forced in Tverberg partitions. The paper constructs counterexamples for every $r\geq 3$, so the assertion is not valid as stated in full generality; the prime-power restriction reflects counterexamples to the topological Tverberg conjecture.