@GhoulishGirl Here's my understanding about how the rumors got started:
Athy would perform her leadership duties and try to help the people from behind the scenes. And she has a good understanding of how economics and everything works, so that people are best helped in the long run, rather than short term fixes.
Her sister would be out in the forefront with the people and soldiers advocating for their well being. And she has a more short-sighted view of how to do things. But because the people see her caring and trying to help, and don't understand the logistics of what Athy has done and was trying to do, they've interpreted that as the older sister being mean and not caring. With Lillian being ill-tempered toward her older sister, it reinforced those thoughts and the rumors circulated.
Lillian isn't a bad person. She has a big heart. But she's immature, close-minded, and unreasonable. Not leadership material. She just thinks she is.
A good way to understand the difference between the two is the fishing anecdote. Lillian is the one who would give a man a fish. Athy is the one who would teach them how to fish. That might be oversimplifying it, but I think you understand what I mean.