Blagojević–Frick–Ziegler's balanced Tverberg partition conjecture

Let $r\geq 2$ be a prime power, $d\geq 1$, and $N\geq (r-1)(d+2)$. Let $k\geq 0$ and $0\leq s<r$ be integers satisfying

$$r(k+1)+s>N+1.$$

For a continuous map $f:\Delta_N\rightarrow\mathbb{R}^d$, where $\Delta_N$ is the $N$-dimensional simplex, consider $r$ pairwise disjoint faces $\sigma_1,\ldots,\sigma_r$.

Blagojević–Frick–Ziegler's conjecture. There are $r$ pairwise disjoint faces $\sigma_1,\ldots,\sigma_r$ of $\Delta_N$ such that

$$f(\sigma_1)\cap\cdots\cap f(\sigma_r)\neq\emptyset,$$

with $\operatorname{dim}\sigma_i\leq k+1$ for $1\leq i\leq s$ and $\operatorname{dim}\sigma_i\leq k$ for $s<i\leq r$.

The paper states that its main theorem proves this conjecture, so the claim is solved rather than open. It is a balanced, dimension-constrained form of the topological Tverberg theorem.