Examples of W$^*$ and C$^*$-superrigid product groups

We provide a new large class $\mathcal C_{AFP}$ of amalgamated free product groups for which the product rigidity result from [CdSS15] holds: if $G_1,\dots,G_n\in\mathcal C_{AFP}$ and $H$ is any group such that $L(G_1\times\dots\times G_n)\cong L(H)$, then there exists a product decomposition $H=H_1\times\dots\times H_n$ such that $L(H_i)$ is stably isomorphic to $L(G_i)$, for any $1\leq i\leq n$. The class $\mathcal C_{AFP}$ contains $W^*$ and $C^*$-superrigid groups from [CD-AD20]. Consequently, we obtain examples of product groups that are both $W^*$ and $C^*$-superrigid.