Yes but that's part of the equilibrium I suppose.

Places where current wages are low are going to see wage increases (because of higher demand / lower supply), which in turn will lead to an increase in costs of living.

Places where current wages are very high are going to see downward pressure on wages (because of lower demand / higher supply), which in turn will eventually lead to a lowering of costs of living.

Obviously that's valid in the "fullness of time". How long it can actually take for this new equilibrium to be fully realized is anyone's guess.