Farah’s conjecture on coanalytic ideals

Prove or refute the conjecture that every coanalytic (Π1_1) ideal on ω is a Farah ideal, i.e., there exists a sequence of hereditary compact sets {K_n}⊆P(ω) such that S∈I iff ∀n ∃k (S\k ∈ K_n).

Background

Farah ideals admit a compact hereditary representation and are known to be Π^0_3 (Fσδ). The paper notes: if every Π1_1 ideal were Farah, then for such ideals either L(I)=Π^0_1 or L(I)=Σ^0_2 would follow, settling the main open question in the coanalytic case.

The authors explicitly cite and rely on the conjecture as a potential route to resolving their dichotomy for L(I).