Here is a question that has been bothering me for some time:
Let G be a finitely generated conjugacy separable group with solvable word problem. Does it follow that the conjugacy problem in G is solvable?
Background. A group G is said to be conjugacy separable is for any two non-conjugate elements x,y in G there is a homomorphism from G to a finite group F such that the images of x and y are not conjugate in F. Equivalently, G is conjugacy separable if each conjugacy class is closed in the profinite topology on G.
A well-known theorem of Mal'cev states that a finitely presented conjugacy separable group has solvable conjugacy problem (in this case it is possible to recursively enumerate all the finite quotients, simultaneously checking conjugacy of the images of two given elements in each of them).
On the first page of the paper 'Conjugacy separability of certain torsion groups.'
(Arch. Math. (Basel) 68 (1997), no. 6, 441--449.) Wilson and Zalesskii claim that the conjugacy problem is solvable in finitely generated recursively presented conjugacy separable groups (which, of course, implies a positive answer to my question), and refer to a work of J. McKinsey, 'The decision problem for some classes of sentences without quantifiers' (J. Symbolic Logic 8, 61 – 76 (1943)). However, I could not find anything in the latter paper that would allow to deal with infinite recursive presentations. Moreover, the corresponding property for residually finite groups simply fails. More precisely, there exist finitely generated residually finite recursively (infinitely!) presented groups with unsolvable word problem (cf. 'A Finitely Generated Residually Finite Group with an Unsolvable Word Problem' by S. Meskin, Proceedings of the American Mathematical Society, Vol. 43, No. 1 (Mar., 1974), pp. 8-10).