Let $\{\mathcal{B}_i\}_{i\in I}$ be a family of sets whose all elements are also sets. If $b,\beta$ are two cardinals such that $$ b\leq \inf\{ \prod_{i \in I} |B_i| : B_i\in \mathcal{B}_i, \mbox{ for all } i\in I\}\leq \sup\{\prod_{i \in I} |B_i| : B_i\in\mathcal{B}_i, \mbox{ for all } i\in I\}\leq \beta. $$ Then, is it true that $$ b\leq \prod_{i \in I}(\inf\{ |B| : B\in \mathcal{B}_i\})\leq \prod_{i \in I}(\sup\{ |B| : B\in \mathcal{B}_i\})\leq \beta ? $$
What about the case that $\leq$ is replaced by $<$ after $b$ and before $\beta$?
Note. If $I$ is finite, then it is true by using this.