An abelian cancellative semigroup embeds (via a semigroup monomorphism) into an abelian group. What about an abelian cancellative Hausdorff topological semigroup that does not embed (via a monomorphism of topological semigroups) into an abelian Hausdorff topological group? And what if the embedding is required to be a homeomorphism when corestricted to the image (endowed with the relative topology induced by the overlying topological group)?

I'm aware of work of E. Schieferdecker (Math. Ann. 131, 1956), N. J. Rothman (Math. Ann. 139, 1960), and a few more authors on the subject, but couldn't find any counterexample in print.