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 can be forced from only a Woodin cardinal (which is provable in ZFC.equiconsistent with the existence of a presaturated ideal and also with the existence of a strong ideal.)