Is the conjugacy problem for two-relator groups known to be undecidable?
The word problem for two-relator groups is a famous open problem (appearing e.g. as Question 9.29 in the Kourovka notebook), and decidability of the conjugacy problem implies decidability of the word problem. However, there are f.p. groups with undecidable conjugacy problem and decidable word problem (one was constructed by Novikov a couple of years before the first known example of a f.p. group with undecidable word problem).
The conjugacy problem for one-relator groups is decidable in the torsion case by the B. B. Newman Spelling Theorem, and remains open in the torsion-free case (see this question and this answer).
It is also known that two-relator groups are not conjugacy separable in general (indeed, one-relator groups need not be conjugacy separable). Furthermore, it is not hard to construct a two-relator group containing a finitely generated subgroup with undecidable conjugacy problem (see e.g. this question). However, decidability of the conjugacy problem is not inherited by subgroups in general, so this does not answer the question.
If no example of a two-relator group with undecidable conjugacy problem is known, then any relevant literature pointing towards either undecidability or decidability of this problem would be appreciated; or indeed any example of a "few"-relator group with undecidable conjugacy problem (the best I know is $12$-relator, which has undecidable word problem).
