The topological Tverberg sunflower conjecture

Let $d\geq 1$, let $r\geq 2$ be a prime power, and set $n=(r-1)(d+1)$. Let $N>n$, and let $\Delta_N$ be the $N$-dimensional simplex. Topological Tverberg sunflower conjecture. For every continuous map $f:\Delta_N\to\mathbb{R}^d$, there exist $r$ pairwise disjoint faces $F_1,\ldots,F_r$ of $\Delta_N$, and an $(n-1)$-dimensional face $F$ of $\Delta_N$, such that for each $n$-dimensional face $F'$ of $\Delta_N$ containing $F$,

$$\bigcap_{i=1}^r f(F_i\cap F')\neq\emptyset.$$

This is posed as an analogue of a Tverberg-sunflower theorem in the topological Tverberg setting, and the supplied source presents it as an open question suggested by Florian Frick.