Equivalence between P?(Σ1_1) and analytic inclusion of L(I)

Determine whether, for every uncountable Polish space X and ideal I on ω, membership I ∈ P?(Σ1_1) (i.e., the existence of an I-scheme A with BI(A)=CI(A)=Q(2^ω)) is equivalent to the inclusion Σ1_1(X) ⊆ L_X(I).

Background

The paper introduces classes of ideals defined via I-schemes A on ω and the sets BI(A), CI(A) ⊆ 2^ω. The class P?(Σ1_1) consists of those ideals admitting an I-scheme with BI(A)=CI(A)=Q(2^ω).

Theorem 9.1 establishes one direction: if I ∈ P?(Σ1_1), then every analytic subset of an uncountable Polish space X belongs to L_X(I). The authors note they do not know whether the converse holds and pose it as an explicit open question.