A solvable conjugacy problem for finitely presented semigroups satisfying C(2) and T(4)

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.