I am aware that the Spelling Theorem of B. B. Newman implies that one-relator groups with torsion are hyperbolic, and thus have a solvable conjugacy problem. My understanding is that for one-relator groups without torsion, the conjugacy problem is still open, though the most recent reference I have for this is over 20 years old.

Has there been any recent developments in this area? Any references for work in this area would be appreciated.

Thanks.