Did the Karaites and Hasidim Essentially Share a Mikveh in the Old City of Jerusalem?

From the Brill entry on the Anan Ben David Synagogue:

“The oldest known Karaite synagogue is the one in Jerusalem. …The northeastern room [of the Karaite Synagogue] served as a ritual bath and may have been connected underground to the ritual bath of the adjacent Tiferet Yisrael Synagogue”.

See my article here for more