The family-stabiliser tree conjecture

Let $G$ be a group and let $\mathcal{F}$ be a family of subgroups.

Family-stabiliser tree conjecture. The inequality $\operatorname{cd}_{\mathcal{F}}G\leqslant 1$ holds if and only if $G$ acts on a tree whose stabilisers generate $\mathcal{F}$.

This folklore conjecture generalises the corresponding result for strictly developable thin simple complexes of groups and would characterise Bredon cohomological dimension at most one through actions on trees. Its resolution status is not specified in the source.