The Bredon Eilenberg–Ganea conjecture

Let $G$ be a group and let $\frakF$ be a semi-full family of subgroups of $G$. Write $\cd_{\frakF}G$ for the Bredon cohomological dimension and $\gd_{\frakF}G$ for the Bredon geometric dimension with respect to $\frakF$. Bredon Eilenberg–Ganea conjecture. One has

$$\cd_{\frakF}G=\gd_{\frakF}G.$$

This generalises the Eilenberg–Ganea conjecture. The equality is established in the source when $\cd_{\frakF}G\geq3$ or $\gd_{\frakF}G\geq4$, while the low-dimensional case remains open.