The maximal embedding dimension criterion for good subsemigroups of $\mathbb{N}^2$

Let $S$ be a good subsemigroup of $\mathbb{N}^2$. Its multiplicity vector is denoted by $\boldsymbol{e}$, and let $M=S\setminus\{\boldsymbol{0}\}$. A good semigroup is said to have maximal embedding dimension when $\operatorname{edim}(S)=e_1+e_2$. Maximal embedding dimension criterion. $S$ is maximal embedding dimension if and only if

$$M+M=\boldsymbol{e}+M.$$

This conjecture extends the analogous characterization for numerical semigroups to good subsemigroups of $\mathbb{N}^2$. The paper establishes that Arf good semigroups have maximal embedding dimension, but the stated equivalence for arbitrary good subsemigroups remains open.