Stability-implies-saturation for density ideals (including Z_0)

Determine whether, for every density ideal I on N (including the asymptotic density-zero ideal Z_0), a reduced product ∏_n M_n/I with stable theory is I-saturated.

Background

Density ideals form a prominent class of analytic F_{σδ} ideals arising from lower semicontinuous submeasures and include the asymptotic density-zero ideal Z_0. The paper proves strong non-saturation results for unstable theories over such ideals but leaves the stable case open.

This problem is a focused instance of the broader question about extending the layered-ideal saturation theorem to wider classes of ideals.