General isomorphism problem for one-relator groups

Determine whether the isomorphism problem is decidable for the class of all one-relator groups, i.e., given two finite one-relator presentations, decide algorithmically whether the resulting groups are isomorphic.

Background

The isomorphism problem for finitely presented groups is undecidable in general, but for many structured classes it can be decided. While there are solutions for particular subclasses of one-relator groups (e.g., torsion or center cases), a complete resolution for all one-relator groups remains elusive.