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 actually ensures 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).