TITLE: Reference request: Recent progress on the conjugacy problem for torsion-free one-relator groups?
QUESTION [9 upvotes]: 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.

REPLY [9 votes]: Yuhasz approach was to prove that every one-relator group satisfies certain version of small cancelation condition and hence has solvable conjugacy problem. Of course if the relator satisfies $C'(1/6)$ the group is hyperbolic and we are done. I think from what he proved the CAT(0)
 conditions $C(3)-T(6)$ and $C(4)-T(4)$ are Ok too but I do not think all cases are covered in his paper. It was long since I looked at his text, so I may forget something.
Victor Guba knows the text well.