co.combinatorics – Counting lattice polytopes by volume

For any $n in mathbb{N}$ and $B in mathbb{R}_{geq 0}$, let $mathcal{P}(n,B)$ be the set of convex polytopes $Delta subseteq mathbb{R}^n$ whose vertices lie in $mathbb{Z}^n$ and whose Lebesgue volume is bounded above by $B$.
It is not difficult to see that the set $mathcal{P}(n,B)$ is finite.

Is there any known asymptotics for the cardinality of $mathcal{P}(n,B)$ as $n to +infty$ and/or $B to +infty$?

Do these asymptotics get more pleasing if we consider the cardinality of $mathcal{P}left(n,frac{B}{n!}right)$, i.e. if we consider the integral volume of polytopes instead of the Lebesgue one?

Are these asymptotics related to the number of simplices with which one can triangulate $Delta in mathcal{P}(n,B)$?

I could not find anything in the literature, but I am not an expert in convex geometry or combinatorics, so I unfortunately do not know the literature very well.
Thank you very much!