My guess says, this. At the very end of the last episode, My Fair Lady-style.
Bonus points if House responds with something along the lines of “Eliza, where the devil are my slippers?”
The difference it makes is he deliberately withheld important information from her for his own selfish motivations. That damages, if not destroys trust and shows her very little respect. House could have grown some balls, said, “Hey, your certificate came in, but I really care about you and don’t want you to leave. Will you stay anyway?” I think she probably would have, but that would mean House would have to show vulnerability and being open to love and he just cannot bring himself to do that. I wouldn’t stay with him after whopper lie like that, but. There is always the possibility, because they dig each other, that she returns to pick up something she left at House’s place, they have a nice long talk about how much he misses her and cares about her, he apologizes for being deceptive, and she agrees to stay anyway. It’s TV; anything could happen.