Pontryagin duality in the class of precompact Abelian groups and the Baire property

We present a wide class of reflexive, precompact, non-compact, Abelian topological groups $G$ determined by three requirements. They must have the Baire property, satisfy the $\mathit{\text{open}}\mathit{\text{refinement}}\mathit{\text{condition}}$, and contain no infinite compact subsets. This combination of properties guarantees that all compact subsets of the dual group $G^{\wedge }$ are finite. We also show that many (non-reflexive) precompact Abelian groups are quotients of reflexive precompact Abelian groups. This includes all precompact almost metrizable groups with the Baire property and their products. Finally, given a compact Abelian group $G$ of weight $\ge 2^{\text{om}}$, we find proper dense subgroups $H_1$ and $H_2$ of $G$ such that $H_1$ is reflexive and pseudocompact, while $H_2$ is non-reflexive and almost metrizable.