Strong jump PA avoidance for the Δ2-Subset principle (D)

Determine whether the Δ2-Subset principle (D) admits strong jump PA avoidance; specifically, ascertain whether for every set X not of PA degree and every set A ⊆ ω (without definability restrictions), there exists an infinite subset Y contained in A or in its complement such that (X ⊕ Y)′ is not of PA degree.

Background

Monin and Patey proved that D admits jump PA avoidance for ∆0,X sets A—namely, for each X ≫ ∅′, every ∆0,X set A has an infinite subset Y in A or its complement with (X ⊕ Y)′ ≫ ∅′.

A stronger version would extend this avoidance to all sets A, not just those ∆0-definable relative to X, and would have consequences for the relationship between Σ1-Subset (equivalently SGST1) and COH.