Never underestimate the power of memory and history to motivate people to act. All the Balkan nations suffered at one time or another at the hands of the Ottomans, all nurse grievances, and some (Bulgaria and Russia, recognizing Russia isn't part of the Balkans) have historic claims to Ottoman territory, stretching back as far as the Byzantine Empire. If the Balkan nations can negotiate with each other, in advance, and lay out what they each are to get, (recognizing there will be lingering issues of distrust) then they would go after the Turks, again, IMO. The biggest issue for them to not act in this way is that they are all still suffering the effects of WWI and before that, the Balkan Wars. Their treasuries are still small and (WAG) still nearly empty and their people tired. If however, Britain decides to bankroll this affair, possibly provide arms and munitions, then its a possibility. Failing to act means the Soviets may intervene on their own. It means the Persian Oil fields the RN needs may be threatened. The big issue in the UK not acting is the war weariness of the people, but if British troops aren't involved directly, this may not matter. I'll leave it to others better versed in British interwar politics to tell more on the plausibility of the above.