Is there a family $\mathcal{P}$ of integral polytopes such that for every $n\in\mathbb N_{>1}$ $\exists p\in\mathcal{P}:vol(p)\in\mathbb N_{>1}$ and

1. $\forall q\in\mathcal{P}$ we have $q\neq p\implies vol(q)\neq n$

3. coordinates of $p$ are $O(\log\log n)$ in bit length

3. dimension of $p$ is $O(\frac{\log n}{\log\log n})$

4. closed in the sense if $vol(p)=ab$ such that $a,b>1\in\mathbb N_{>1}$ implies $p=p_1\star p_2$ for a polytope product $\star$ where $vol(p_1)=a$ and $vol(p_2)=b$?

Is there any way to describe such a family so that given $n\in\mathbb N_{>1}$

4. $p$ cannot be described in polynomial time from $n$?