Dichotomy for coanalytic ideals: closed or Fσ?

Ascertain whether, for every coanalytic (Π1_1) ideal I on ω and every uncountable Polish space X, the family L_X(I) must be either the closed sets Π^0_1(X) or the Fσ sets Σ^0_2(X).

Background

The authors observe a heuristic that the topological complexity of L(I) tends to be lower than the descriptive complexity of the ideal I itself. They then conjecture a two-way dichotomy for coanalytic ideals.

They note that, for Farah ideals, either L(I)=Π^0_1 or L(I)=Σ^0_2 is known, and that a positive resolution of Farah’s conjecture would imply a positive answer here.