Random groups contain subgroups commensurable with orbifold geodesic-spine groups

A geodesic spine $K$ is a finite $2$-dimensional complex with totally geodesic edges and faces that embeds in a hyperbolic $3$-manifold $M$ with totally geodesic boundary as a deformation retract. An orbifolding $K'$ of $K$ is a $2$-dimensional orbifold complex whose underlying complex is homeomorphic to $K$ and which has at most one orbifold point added to each face. Let $G$ be a random group, either in the few-relators model or in the density model with density $0<D<1/2$.

Random-spine subgroup conjecture. For any fixed geodesic spine $K$, with overwhelming probability $G$ contains a subgroup commensurable with the orbifold fundamental group of some orbifolding $K'$ of $K$.

This conjecture predicts that random groups contain subgroups reflecting the controlled combinatorial structure of thin geodesic spines; the source provides no evidence of a resolution, so its status remains open.