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.
Girl please don't get all flustered by the fact that he's gay no one asked to fell for him he don't need your useless love anyway