General conjugacy problem for one-relator groups

Determine whether the conjugacy problem is decidable for all one-relator groups given by a finite presentation with a single defining relator.

Background

Dehn posed the conjugacy problem as one of the three fundamental decision problems for finitely presented groups. Magnus solved the word problem for all one-relator groups, but the status of the conjugacy problem has remained unresolved in full generality despite many partial results (e.g., for hyperbolic, torsion, cyclically pinched, and some other subclasses).