We knew from the start that Olivia killed her husband the Prince in the "story" version of events. So I'm guessing that the implication is that Olivia has been spreading the rumours about the potential adulterous nature of the Prince and the head of house Lachia, to get some use out of him before she gets rid of him. Set herself up as victim of an affair who choose to side with the Lachia family.
However, everyone miscalculated. They all assume that Ran and Yustav must be having a power struggle behind the scenes regarding who keeps control of the Duchy. In reality Ran fully intends to just give it to Yustav, and Yustav probably intends to propose to he after the fact. They're not blood related, so if he doesn't acknowledge her as family and then strips her of her familial status/ ties, he is then free to marry her.