I don't get what the issue is?
Will FMC lose friends or become isolated if she's 'dating' someone? No, obviously not.
Was she previously in a relationship that she now can't enter because of the rumour? Nope.
This rumour is nothing that can't be ironed out whenever FMC or MC talk to people, there's no reason to rush to solve it, or dwell on it for an entire chapter.