The extensional ESO NP-intermediate problem conjecture

An extensional ESO sentence $\Phi$ defines a decision problem by asking whether the sentence holds on the input structure. Extensional ESO NP-intermediate conjecture. There is an extensional ESO sentence $\Phi$ such that deciding $\Phi$ is NP-intermediate. The surrounding discussion points to examples polynomial-time equivalent to Graph Isomorphism and to the complement of Monotone Dualization, but does not establish the existence of an NP-intermediate problem; this conjecture remains open.