Comparative strength of Second-order Absolute Morley variants

Determine whether the principle stating that if E is an equivalence relation on R that is a countable intersection of projective sets and there is no perfect set of pairwise E-inequivalent reals, then E has at most aleph_1 equivalence classes, is strictly weaker than the corresponding principle for sigma-projective equivalence relations.

Background

The authors introduce two formulations of an absolute Morley-type dichotomy for equivalence relations on the reals: one for sigma-projective equivalence relations and a seemingly weaker version for equivalence relations that are countable intersections of projective sets.

Following prior work, the sigma-projective version implies the general Second-order Absolute Morley. The authors explicitly note that it is unknown whether the countable-intersection variant is strictly weaker than the sigma-projective variant.