Subgroup dichotomy for cyclically hyperbolic cubulable groups

Let $G$ be a group acting geometrically on a CAT(0) cube complex. A group is cyclically hyperbolic if it admits a non-elementary acylindrical action on a hyperbolic space with a loxodromic element whose centralizer is virtually cyclic. A group is SQ-universal if every countable group embeds in a quotient of it. Subgroup dichotomy conjecture. If $G$ is cyclically hyperbolic, then every subgroup of $G$ is either virtually abelian or SQ-universal.

This is expected to strengthen the paper's subgroup alternative, generalising the corresponding result for $G$ itself. The conjecture is presented as an expectation, and no resolution is given in the supplied text.