The conjugacy problem in finitely generated subdirect products of torsion-free hyperbolic groups is solvable if and only if cyclic subgroup membership is uniformly decidable in G1/(P ∩ G1).
Collins, The word, power and order problems in finitely presented groups,in ”Word Problems, Deci- sion Problems and the Burnside Problem in Group Theory” (W.W
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On the conjugacy problem for subdirect products of hyperbolic groups
The conjugacy problem in finitely generated subdirect products of torsion-free hyperbolic groups is solvable if and only if cyclic subgroup membership is uniformly decidable in G1/(P ∩ G1).