The Baire property and precompact duality

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We prove that if G G is a totally bounded Abelian group such that its dual group G ^ p \widehat {G}_p equipped with the finite-open topology is a Baire group, then every compact subset of G G must be finite. This solves an open question by Chasco, Domínguez, and Tkachenko. Among other consequences, we obtain an example of a group that is g g -dense in its completion but is not g g -barreled. This solves a question proposed by Außenhofer and Dikranjan.