Necessary and sufficient condition for exact hereditary Δ_n-conservativity

Develop a necessary and sufficient condition on a pair (T, U) of computable consistent extensions of Peano Arithmetic that is equivalent to the existence, for some n ≥ 1 and some formula class Θ with Θ ⊇ Σ_n, of a sentence φ in Θ that is simultaneously exactly hereditarily Δ_n-conservative over T and U. Specifically, require that φ ∈ HCons(Δ_n, T) ∩ HCons(Δ_n, U) and that for every class Θ′ with Θ′ ⊄ Δ_n there exists a Θ′ sentence ψ such that PA + φ ⊢ ψ while T ⊬ ψ and U ⊬ ψ.

Background

The paper studies a variety of partial conservativity notions over pairs of theories T and U, including Σ_n-, Π_n-, Δ_n-, Σ_n ∧ Π_n-, and 𝔅(Σ_n)-conservativity and their hereditary and exact variants.

For the Δn row of Table 1 (items marked as (i)), the authors show that C_n (both T + Th{Πn}(U) and U + Th{Πn}(T) consistent) is a sufficient condition and C_n (both T + Th{Σn ∧ Π_n}(U) and U + Th{Σ_n ∧ Π_n}(T) consistent) is a necessary condition (Theorem 6.1: Thm_D_H). However, a full necessary-and-sufficient characterization for each Θ ⊇ Σ_n remains unsettled.