Responses
It's not expressly stated but I think they did it. At the beginning the nanny/maid of the other one asks if there was a problem during the emperor's last visit to her quarters. So I think they did it.
You can have love without sex and sex without love, specially if it's their duty.
Did they do the deed? I mean I think they did or else she wouldn't tell such obvious lie the ml is not stupid he would know if he had sex