This is the first time that the Lady actually got pissed. And the thing about how the Moore's have the ability to see through lies makes me think that she likes Rune because of how the Lady sees Rune as a pure and honest soul.
If I could see through lies, I would always find it hard to really connect with people intimately because I would know that they are able to lie so easily, but if I were to see people that are just honest, I would want to connect with them.
Either way, I feel that Rune should explain that she knows who has the necklace now. Sucks how Kaito revealed himself when things were at their worst for him.