Suppose I have a volatility smile for a certain underlying at a given maturity. This implies a certain density for the underlying at that maturity, which can be explicitly computed, via a corollary to Dupire's formula.
Now suppose that I bump up the - say - right wing of the smile, i.e. all implied vols stay the same from $ K= 0 $ to $ K= F_t$, the forward, while the rest of the smile is bumped up a little.
What happens to the implied density of the underlying?
My reasoning was: implied vol goes up => call prices go up => higher probability that underlying price is higher => fatter right tail.
But from the put-call parity, also put prices go up => fatter left tail.
So, I would conclude that both tails increase, but this is at odds with the corollary to Dupire's formula, which is local, i.e. a bump in the right wing only changes derivatives there, so it would leave the left tail untouched.
What is wrong with my reasoning then?