That suspicion check actually checks your suspicion stat (visible under the character's name in the Grimoire). Your suspicion has to be under a certain value (15 for adventure mode, I believe) to pass it.
The suspicion stat is something you raise sometimes with certain choices if it doesn't fit your public behaviour when someone in the High Council is present. (For example, choosing a bloodthirsty answer while your character is overly considerate and charismatic).
That said, the suspicion check in that week doesn't need to be passed. It will decrease your faction approvals a bit, but you can easily get those back with tea parties or later checks.