On the supremum of the pseudocompact group topologies

P is the class of pseudocompact Hausdorff topological groups, and P′ is the class of groups which admit a topology T such that (G,T)∈P. It is known that every G=(G,T)∈P is totally bounded, so for G∈P′ the supremum T∨(G) of all pseudocompact group topologies on G and the supremum T#(G) of all totally bounded group topologies on G satisfy T∨⊆T#.The authors conjecture for abelian G∈P′ that T∨=T#. That equality is established here for abelian G∈P′ with any of these (overlapping) properties. (a) G is a torsion group; (b) |G|⩽c2; (c) r0(G)=|G|=ω|G|; (d) |G| is a strong limit cardinal, and r0(G)=|G|; (e) some topology T with (G,T)∈P satisfies w(G,T)⩽c; (f) some pseudocompact group topology on G is metrizable; (g) G admits a compact group topology, and r0(G)=|G|. Furthermore, the product of finitely many abelian G∈P′, each with the property T∨(G)=T#(G), has the same property.