Betti-element count distinguishes finite abelian groups via block monoids

Let $G$ be a finite abelian group, and let $\mathcal{F}(G)$ be the free abelian monoid on $G$. Define

$$\operatorname{eval}\left(\prod_{g\in G}g^{\alpha_g}\right)=\sum_{g\in G}\alpha_g g$$

and let

$$\mathcal{B}(G)=\left\{x\in\mathcal{F}(G):\operatorname{eval}(x)=0\right\}$$

be the block monoid on $G$. The Betti elements of an affine monoid are the elements whose factorization graph is disconnected. Betti-element count conjecture. Let $G$ and $G'$ be finite abelian groups of order greater than $2$. Apart from $\mathbb{Z}_3$ and $\mathbb{Z}_2\oplus\mathbb{Z}_2$, if $G\not\cong G'$, then the numbers of Betti elements of $\mathcal{B}(G)$ and $\mathcal{B}(G')$ are different. Computations suggest that the number of Betti elements grows rapidly with the size of $G$, but the conjectured distinction between all nonisomorphic groups beyond the stated exceptions remains unproved.