WI: Switzerland attacked during World War 2

kernals12

Banned
Switzerland, famously, avoided both World Wars. But there was a plan drawn up for an invasion called Operation Tannenbaum. Switzerland is not impossible to invade, the French did it in 1798. But Berlin decided against it thinking it wasn't worth a brutal mountain battle for a strategically unimportant bit of land even if Hitler loathed the Swiss and their democratic ways.

But even if such an invasion was unsuccessful, it would've certainly resulted in many deaths and tremendous damage.

My question is: Would this convince the Swiss to abandon their neutrality and join NATO after the war?
 
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