What if Rum was the Ottoman Empire of this world?

In an alternate universe after 1,000 years of life, the flag of Rum waves victoriously over Constantinople. Rum, once a small sultanate in southern Anatolia, has grown to dominate the region. now the question is, what's next for rum? who will they invade next? will they stray from their Greek culture or embrace it? you guys discuss it down below.
Name-wise, they're at least an easier transition! The Ruman Empire.
and after it transitions out of being the Rum Empire it can be called Rumania! ...wait
jk--as chance would have it, i'm actually also using the Rum as the replacement Ottomans for my ASB ATL and, after weighing other options, i decided "Rumania" is probably the most likely name for them post-Empire if it wasn't just called "Turkey". also irony since Romania isn't a thing in this same TL