Equality of the three-dimensional abelian Cremona group classes

Let $\mathcal{A}_3$ and $\mathcal{B}_3$ be the classes of finite abelian groups arising in the paper, and let

$$\mathcal{A}'_3=\mathcal{PA}_3=\mathcal{A}_1\times\mathcal{A}_2,$$

while

$$\mathcal{B}'_3=\mathcal{A}_3+\{G_1,G_2,G_3,G_4\},$$

where $G_1=(\mathbb{Z}/4)^4$, $G_2=(\mathbb{Z}/8)^2\times(\mathbb{Z}/4)\times\mathbb{Z}/2$, $G_3=(\mathbb{Z}/6)^2\times(\mathbb{Z}/3)^2$, and $G_4=(\mathbb{Z}/6)^3\times\mathbb{Z}/2$. The inclusions $\mathcal{A}'_3\subseteq\mathcal{A}_3$ and $\mathcal{B}'_3\subseteq\mathcal{B}_3$ hold. Three-dimensional class conjecture. One has

$$\mathcal{A}_3=\mathcal{A}'_3,\qquad \mathcal{B}_3=\mathcal{B}'_3.$$

The conjecture would classify the relevant finite abelian groups in dimension three by showing that no groups beyond the explicitly listed candidates occur. The supplied context does not state whether these equalities have been proved or disproved.