Strength of the quantified here-and-there logic with constant domains (HTCD)

Determine the exact strength of the intermediate predicate logic HTCD = IQC + HTQ + CD by locating it within the lattice of intermediate predicate logics between IQC and CQC, and ascertain whether HTCD plays an extremal role (e.g., being the strongest proper intermediate predicate logic) analogous to the propositional logic HT.

Background

The paper considers the first-order logic HTCD obtained by adding the quantified here-and-there axiom (HTQ) and the constant domain axiom (CD) to intuitionistic first-order logic (IQC). It recalls known results: HT is the strongest proper intermediate propositional logic, and HTCD is sound and complete with respect to 2-node intuitionistic models with constant domains.

However, the author explicitly states that the exact strength of HTCD—its position among intermediate predicate logics and whether it mirrors the extremal property of HT at the propositional level—remains unknown.