Faithfulness conjecture for laminations arising from group actions

Let $\mathcal L$ be a lamination arising from a group action. A fidelity of $\mathcal L$ is a germ map into a path-connected space whose induced map on fundamental germ groupoids is a groupoid monomorphism, and $\mathcal L$ is faithful if it has a fidelity. Faithfulness conjecture. Every lamination arising from a group action is faithful. Suspensions are known to be faithful, and the inclusion of the linear foliation of a torus induced by a plane is a fidelity; the general existence of fidelities is described as interesting but difficult and remains open.