Existence of a Σ_n sentence for a specific pair (T, U) yielding non-trivial hereditary Δ_n-conservativity

Determine whether there exists, for a fixed n ≥ 1, a Σ_n sentence ψ that is simultaneously non-trivially hereditarily Δ_n-conservative over T := PA + ¬Pr_{PA}^{Σ_n}({0=1}) and U := PA + Pr_{PA}^{Σ_n}({0=1}); i.e., ψ ∈ HCons(Δ_n, T) ∩ HCons(Δ_n, U) and PA ⊬ ψ, T ⊬ ψ, and U ⊬ ψ.

Background

The authors examine a concrete pair of theories: T = PA + ¬Pr_{PA}^{Σ_n}(0=1) and U = PA + Pr_{PA}^{Σ_n}(0=1). They note that (T, U) is a C_n-pair but not a C_n-pair, and Theorem 6.2 (Thm_D_ntH) yields a Σ_n sentence φ with strong Δ_n(PA)-conservativity properties over subtheories of T and U.

Despite this, it remains unknown whether there is a Σ_n sentence that is simultaneously non-trivially hereditarily Δ_n-conservative over this particular pair.