Realizing specific Borel classes as L(I)

Determine whether there exist an ideal I on ω and a positive integer α such that, for an uncountable Polish space X, the family of sets of I-limit points satisfies L_X(I) = Σ^0_{2α−1}(X) or L_X(I) = Π^0_{2α}(X).

Background

Earlier results show strong constraints on which Borel classes can equal L(I): for example, L(I) cannot be E_1 (open sets), Π^0_2 (Gδ), or Σ^0_3, while some high-complexity families (e.g., Π^0_{2α−1} and Σ^0_{2α} obtained from certain Fubini products) can occur as inclusions.

This question asks for exact realizations of entire Borel classes at alternating parity levels—whether some ideal I can produce L(I) equal to Σ^0_{2α−1} or Π^0_{2α}.