So like the Luteces totally love Elizabeth and Robert and Rosalind decide to interfere a bit more than they already have and try to convince Atlas to not kill her. To get cozy with him, they decide Robert must form some kind of physical/emotional connection (since Atlas is obv. gay).

tl;dr: Robert Lutece seduces Atlas into not killing Elizabeth and everyone lives happily ever after yayyyyyyyyyyyyyyyyy sex solves problems