Stability implies saturation for ideals with non-aleph-1-saturated quotients?

Determine whether, for an ideal I on N such that the quotient Boolean algebra P(N)/I is not aleph_1-saturated, every reduced product ∏_n M_n/I whose first-order theory is stable is nevertheless I-saturated.

Background

The main saturation theorem establishes that for layered ideals, stable reduced products are maximally saturated. The authors ask whether this conclusion can extend beyond layered ideals to those with non-aleph_1-saturated quotients.

They note that the answer may depend on the specific first-order theories involved, indicating a nuanced interaction between set-theoretic properties of the ideal and model-theoretic stability.