Treaty of Sevres ratified in 1920- Does Turkey join the Axis in WW2?

Hello, first thread here.
As the title says. A lot of questions come to mind: Would Turkey intervene in the Italian invasion of Greece? Would Turkey let the Germans pass through their territory to invade the Middle East? Could logistics allow any kind of invasion? What would be the status of Armenia, Soviet or independent? Would Turkey invade the USSR in the Caucasus? (I'd say this one is pretty unlikely)
Here is a map of how Turkey would've approximately looked like for reference.

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