Grigorchuk's stronger conjecture for finitely presented groups

Let $G$ be a finitely presented group. A free subsemigroup on two generators is a subsemigroup generated by two elements with no nontrivial relations. Grigorchuk's stronger conjecture. Either $G$ contains a free subsemigroup on two generators or $G$ is virtually nilpotent. This strengthens the conjecture that finitely presented groups have no intermediate growth; the latter is described as a main remaining open problem, and the stronger statement is also open.