Common cubulable quotients of acylindrically hyperbolic groups

A group is cubulable if it acts properly and cocompactly on a median graph, and it is acylindrically hyperbolic if it admits a non-elementary acylindrical action on a hyperbolic space. Common quotient conjecture. Any two acylindrically hyperbolic cubulable groups admit a common quotient that is itself acylindrically hyperbolic and cubulable. This proposes a permanence and common-quotient phenomenon for the class of acylindrically hyperbolic cubulable groups; it is stated in the paper as an open direction for future work.