The Baire property and precompact duality

We prove that if G is a totally bounded abelian group \ its dual group Gₚ equipped with the finite-open topology is a Baire group, then every compact subset of G must be finite. This solves an open question by Chasco, Dom\'inguez and Tkachenko. Among other consequences, we obtain an example of a group that is g-dense in its completion but is not g-barrelled. This solves a question proposed by Au{enhofer and Dikranjan. }