NPR ran a story on the restoration of the Studios monastery and its conversion into a mosque. They also mentioned the similar plans for Hagia Sophia and the conversion of churches into mosques already underway in Trabzon and Iznik (Nicaea).
The Turkish Muslim-lead nation-state is primarily concerned with the history of its land and peoples after 1453 which is why they emphasize the more recent history of their architectural monuments and heritage. NPR’s story focuses on the loss to the Greek community living in Istanbul if these churches were turned into mosques, but the impact of this conversion will reach much further beyond that city and that community.
UPDATE: The Studios Monastery was founded in 462. It was once the largest monastery in Constantinople and the cloister was believed to hold as many as 700 monks at a time. It was destroyed by the Crusaders in 1204 during the Fourth Crusade, but then rebuilt in 1290 by Constantine Palaiologos after Byzantine Greek rulers from Nicaea overthrew the Latin Kingdom of Constantinople. It was again largely destroyed in 1453 by the Ottoman Turks. The stones from the monastery ruins are believed to have been used by neighbouring residents to repair their homes and the thirteenth century pavement remained exposed to the open air. The remains of the cathedral dedicated to St. John the Baptist survive as probably the oldest remaining church in Constantinople. The monastery’s remains have been accessible as a museum since 1946. Wikimedia Commons has some nice views of the exterior and more images of some well-known manuscripts produced there.