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?