The original Eilenberg-Ganea conjecture can be formulated as: any group of cohomological dimension 2 has geometric dimension 2. It remains unsettled.

On the other hand, various generalisations in which the ordinary cohomological (resp. geometric) dimensions are replaced with the cohomological (resp. geometric) dimensions with respect to a certain family of subgroups are known to be false. Examples of groups $G$ for which $\operatorname{cd}_\mathcal{F}(G)=2$ and $\operatorname{gd}_\mathcal{F}(G)=3$ have been found by Brady, Leary and Nucinkis when $\mathcal{F}$ is the family of finite subgroups, and by Fluch and Leary when $\mathcal{F}$ is the family of virtually cyclic subgroups.

This got me wondering about positive about positive Eilenberg-Ganea results.

Any class of groups which is closed under isomorphism and taking subgroups can be used to define a family of subgroups of any given group. One can then ask if the Eilenberg-Ganea conjecture holds for this particular family. For the trivial class this is unsettled; for the class of all groups the result is vacuously true because every group has cohomological and geometric dimension zero with respect to the family of all subgroups. Thus the question becomes:

Does there exist a (proper) class $\mathcal{X}$ of groups such that for $\mathcal{F}$ the family of subgroups of a given group consisting of groups in $\mathcal{X}$, every group $G$ such that $\operatorname{cd}_\mathcal{F}(G)=2$ has $\operatorname{gd}_\mathcal{F}(G)=2$?