It is consistent that the nonstationary ideal on $\omega_1$ is strong but not pre-saturated.  Baumgartner and Taylor proved in the aforementioned paper that strong ideals are preserved by c.c.c. forcing and asked whether the same is true for pre-saturated ideals.  The answer to this question is negative, implying a negative answer to the question I posted above.  Apparently this was first proved by Veličković in the paper _Forcing axioms and stationary sets_ (which I cannot seem to access online) from ZFC + SPFA.  Another example of a c.c.c. forcing that destroys pre-saturation may be found in a more recent paper by Larson and Yorioka, _Another c.c.c. forcing that destroys presaturation_, assuming the consistency of ZF + AD.

I don't know if a negative answer is provable in ZFC.