Pseudocompact totally dense subgroups

It was shown by Dikranjan and Shakhmatov in 1992 that if a compact abelian group K admits a proper totally dense pseudocompact subgroup, then K cannot have a torsion closed G_delta-subgroup; moreover this condition was shown to be also sufficient under LH. We prove in ZFC that this condition ensures actually the existence of a proper totally dense subgroup H of K that contains an omega-bounded dense subgroup of K (such an H is necessarily pseudocompact). This answers two questions posed by Dikranjan and Shakhmatov (Proc. Amer. Math. Soc. 114 (1992), 1119-1129)..