Decidability conjecture for deficiency-one metabelian groups

Let $G$ be a metabelian group given by a full-rank presentation

$$G=\langle A\mid R\rangle_{\mathcal{M}}.$$

Deficiency-one decidability conjecture. If $|A|-|R|=1$, then the Diophantine problem in $G$ is decidable.

The cases of deficiency at most $-2$ and at least $0$ are settled by the preceding theorem, while deficiency one remains an open case. Some deficiency-one presentations define groups with decidable Diophantine problem, including certain Baumslag–Solitar groups and cyclic groups, but the general case is unresolved.