Elementary embedding into the HNN extension over a finite subgroup

Prove that for a non-elementary hyperbolic group G and a finite subgroup C such that N_G(C) is non-elementary and C is the maximal finite subgroup normalized by N_G(C), the inclusion G ↪ G′ defined by the HNN extension G′ = ⟨G, t | tg = gt for all g ∈ C⟩ is an elementary embedding (not merely an EAE-embedding).

Background

Theorem \ref{EAE} shows that under the stated hypotheses the inclusion G ↪ G′ is an EAE-embedding. The authors conjecture a stronger result: that the embedding is elementary. Achieving this would require quantifier elimination tools beyond current results (which are available in the torsion-free case only).

Establishing elementary embedding here would provide a powerful extension principle for hyperbolic groups with torsion and clarify the model-theoretic behavior of HNN extensions over finite subgroups.