Use DDNS_C instead of DNS^forall_C in constructive completeness with disjunction

Determine whether the constructive completeness proof for classical first-order logic with respect to classical possibly-exploding Tarski semantics in the presence of disjunction can be carried out using only the generalised disjunctive Double Negation Shift principle DDNS_C, instead of requiring the generalised Double Negation Shift DNS^forall_C for exists-free formulas; equivalently, either derive DNS^forall_C for exists-free formulas from DDNS_C or develop a modified completeness proof that depends solely on DDNS_C. Here DNS^forall_C denotes the schema ∀n (((A(n) → C) → C) → (((∀n A(n)) → C) → C)) with C a Σ^0_1-formula, and DDNS_C denotes the generalised disjunctive double-negation shift schema ∀n ¬_C(∀D ¬_D(¬_D A(n) ∨ ¬_D B(n))) → ¬_C(∀D ¬_D ∀n (¬_D A(n) ∨ ¬_D B(n))), where ¬_E X abbreviates (X → E).

Background

In their constructive analysis of Henkin's proof, the authors extend completeness to include disjunction by employing a generalised Double Negation Shift principle DNS^forall_C for Σ^0_1 formulas, in addition to possibly-exploding Tarski semantics. This extension requires managing double negation across quantifiers and disjunction in the meta-theory.

They discuss recent related results suggesting that a weaker disjunctive principle, DDNS_C, may suffice to handle disjunction constructively in related logical settings (e.g., work by Kirst on propositional epistemic logic). Motivated by this, they explicitly conjecture that their completeness argument could be based on DDNS_C rather than the stronger DNS^forall_C, either by deriving the latter (restricted to exists-free formulas) from DDNS_C or by revising the proof to rely solely on DDNS_C.