Could Constantinople have fallen during the First Balkan War?

With more Greek spending and more Ottoman corruption, could Constantinople have fallen in 1913?

Possibly: Greek control of the straits could have cut Ottoman reinforcement while Bulgarian armies closed on the city, making Bulgaria the likelier captor.

The counterfactual depends on Greece defeating the Ottoman fleet and forcing the Dardanelles despite the coastal guns. Bulgarian forces had already driven the opposing Ottoman army deep into Gallipoli, so a Greek breakthrough could have gained allied support on one bank of the straits. Greek sea power could then have cut off reinforcements reaching Constantinople from the Asian side while Bulgarian armies pressed from Europe. This combination might have made the city fall, although the Bulgarian rather than Greek army was geographically positioned to take it.