There are multiple, inequivalent, definitions for conjugacy in semigroups. In Cummings and Jackson (Semigroup Forum 88, 52-66, 2014), we conjectured that, for at least one of these definitions of conjugacy, the conjugacy problem for finitely presented semigroups satisfying C(2) and T (4) is solvable. Here we essentially verify that conjecture. In that 2014 Semigroup Forum publication, we developed geometric methods to solve a conjugacy problem for finitely presented semigroups satisfying C(3). We use those methods again here.