Computable d-$\Sigma_2$ Scott sentences for finitely presented groups

Let $G$ be a finitely presented computable group. A computable $d$-$\Sigma_2$ Scott sentence is a Scott sentence for $G$ that is the conjunction of a computable $\Sigma_2$ sentence and a computable $\Pi_2$ sentence. Finitely presented group Scott sentence conjecture. Every finitely presented computable group $G$ has a computable $d$-$\Sigma_2$ Scott sentence. This question arose because a finitely generated computable group is known to have a computable $\Sigma_3$ Scott sentence that is optimal, while the known counterexample is not finitely presented; the paper describes evidence for a positive answer but leaves the finitely presented case open.