Mutual equivalence of the Δ_n exact hereditary items (i) in Table 1

Determine whether the items marked (i) in Table 1—each asserting the existence, for some n ≥ 1 and various choices of Θ ⊇ Σ_n, of a sentence that is simultaneously exactly hereditarily Δ_n-conservative over T and U—are mutually equivalent; that is, ascertain whether the existence for one such Θ implies the existence for the others among Θ ∈ {Σ_n, Σ_n ∧ Π_n, 𝔅(Σ_n), Δ_{n+1}(PA)}.

Background

In Table 1 the authors report their strongest results for simultaneous exact hereditary Γ-conservativity. For Γ = Δ_n the entries are marked (i), reflecting that C_n is sufficient and C_n is necessary (Theorem 6.1), but a precise necessary-and-sufficient condition for each Θ ⊇ Σ_n is unresolved.

Beyond individual characterizations, it is unknown whether these (i) items across different choices of Θ are logically equivalent conditions on T and U.