Non-recursive enumerability of the minimum equational graph theory

A graph model is a graph model of the $bb$-calculus, and its equational theory is denoted by $Th(\mathscr{M})$. The minimum equational graph theory is the least equational theory among graph models.

Minimum-theory conjecture. The minimum equational graph theory is non-r.e.

The paper states that it remains open whether the minimum equational graph theory is r.e.; this is the weakest of the listed open instances of the Scott-semantics conjecture.