Nucinkis's conjecture on finite-dimensional classifying spaces for proper actions

Let $G$ be a group, and let $\operatorname{\mathfrak{F}}\operatorname{cd}G$ denote its cohomological dimension with respect to the family $\operatorname{\mathfrak{F}}$ of finite subgroups. Write $\operatorname{E}_{\operatorname{\mathfrak{F}}}G$ for the classifying space for proper actions.

Nucinkis's conjecture. Every group of finite $\operatorname{\mathfrak{F}}$-cohomological dimension admits a finite-dimensional model for $\operatorname{E}_{\operatorname{\mathfrak{F}}}G$.

The conjecture is motivated by bounds relating Bredon cohomological invariants to finite-dimensional models for proper actions. The source presents it as an open conjecture and attributes it to the second author.