This is one of many open questions in geometric group theory related to quasi-isometries. Proving things about invariance under quasi-isometries is generically quite tricky, as quasi-isometries do not even need to be continuous. Some other open questions:
- Is the Haagerup property invariant under quasi-isometries? (see comment by YCor for recent work on this one)
- Is the rapid decay property invariant under quasi-isometries?
- Is the property of having uniform exponential growth invariant under quasi-isometries?
- Are random finitely presented groups quasi-isometry rigid?
- How can fundamental groups of compact $3$-manifolds be classified up to quasi-isometry?