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).
Ol’shanskii, On residualing homomorphisms and G-subgroups of hyperbolic groups , In- tern
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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).