The exact formula conjecture for the topological Tverberg threshold

Let $N_r(d)$ denote the least integer $N$ such that every continuous map from the $N$-simplex to $\mathbb{R}^d$ identifies points from $r$ pairwise disjoint faces. The parameters satisfy $r\ge 2$ and $d\ge 1$.

Exact formula conjecture.

$$N_r(d)=\begin{cases}(r-1)(d+1)&\text{if }r\text{ is a power of a prime or }d\le r,\r(d+1)-1&\text{otherwise}.\end{cases}$$

This predicts the precise threshold governing the failure of the topological Tverberg phenomenon. The supplied text does not provide evidence resolving the conjecture.