The latter is correct.
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addyaddyI suggested that he initiates conversations with family members.[I hinted / I led them to believe] that he regularly initiates conversations with family members.
addyaddyI suggested that he initiate conversations with family members.I [made the suggestion / advised hi
CalifJim[I hinted / I led them to believe] that he regularly initiates conversations with family members.Aha! I thought of this, too. I knew I wasn't losing my mind!
I don't think you intend this meaning.
Aspara GusI knew I wasn't losing my mind!Well, that's a good feeling, isn't it?
CalifJim[I hinted / I led them to believe] that he regularly initiates conversations with family members.In BrE that can be "I suggested that he initiates/initiated ..."
CalifJimI [made the suggestion / advised him] that [he should / it would be a good idea for him to] initiate conversations with family members.That can be
CalifJim addyaddyI suggested that he initiates conversations with family members.[I hinted / I led them to believe] that he regularly initiates conversations with family members.I don't think you intend this meaning.addyaddyI suggested that he initiate conversations with family members.I [made the suggestion / advised him] that [he should / it would be a good idea for him