Necessary and sufficient condition for Σ_n non-trivial 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, of a Σ_n sentence φ that is simultaneously non-trivially hereditarily Δ_n-conservative over T and U; i.e., φ ∈ HCons(Δ_n, T) ∩ HCons(Δ_n, U) and φ ∉ Th(T) ∪ Th(U).

Background

Table 2 addresses the existence of sentences that are simultaneously non-trivially hereditarily Γ-conservative over T and U. For the case Γ = Δn with Θ = Σ_n (marked as (ii)), the authors show that C_n (at least one of T + Th{Πn}(U) or U + Th{Πn}(T) consistent) is a sufficient condition and C_n (both T + Th{Πn}(U) and U + Th{Π_n}(T) consistent) is a necessary condition (Theorem 6.2: Thm_D_ntH).

Despite these partial results, the authors have not identified a single condition that is both necessary and sufficient for the Σ_n case.