An interesting question that has an interesting answer, because the Ottoman Turks never had the Byzantine crown jewels, due to the fact that they'd been pawned to Venice in 1343 by Anna of Savoy.

Anna had been the wife of Emperor Andronikos III, but he died in 1341 and his young son John V was nine years old, so Anna assumed the regency. The government administration, however, was in control of a John Kantakouzenos, and mutual distrust and intrigue between him and Anna would eventually lead to civil war. In an attempt to drum up pro-Western support (since Anna had little to no native Greek allies due to their distrust of Westerners) and pay for an army, she pawned off the crown jewels to the Republic of Venice for 30,000 Venetian ducats. This ended up being a futile gesture and she lost the war. John V and multiple successive emperors of Byzantium would continually attempt to negotiate a return of the jewels, but Venice would never end up letting them go until and after Byzantium's fall to the Ottomans. I draw this information from John Julius Norwich's "Byzantium" trilogy.

Further reading online details that Napoleon's conquest of Venice would lead to many pieces of the original set being looted and melted down to finance his wars, although a good amount of the crown jewel set can still be found in Venice, as well as other sets given away as gifts in Hungary and Mt. Athos.