Non-recursive enumerability for effective graph models

Let $\mathscr{M}$ be an effective graph model, that is, a graph model satisfying the paper's effectiveness condition. Write $Th(\mathscr{M})$ for its equational theory.

Effective graph-model conjecture. All the effective graph models have non-r.e. equational theories.

This is stated as an open instance of the Scott-semantics conjecture, stronger than the claim about the minimum equational graph theory. The surrounding results establish non-r.e. order theories for all graph models, but not this equational-theory claim.