In one chapter (I think it was chapter 22 of the manhwa), Elliot asked Lillian about her opinion on genes and hereditary personality traits and then talked about their similarities/differences, so, for me, it sounded like they were kind of in a situation similar to the one between Elliot and William - it's incestuous to have feelings for a sibling and Elliott couldn't cope with William's feelings, so maybe it was something like "does Lillian share that kind of tendency with William and would she (be able to) fall in love with me as well?".
(Besides Elliott wanting to fulfill William's wish, of course.)