Enumerative characterisation of boolean lattices II

Question

This is a sequel of this post.

The boolean lattice $B_n$ is graded with rank numbers $\binom{n}{0}, \binom{n}{1}, \dots, \binom{n}{n}$, and $n2^{n-1}$ edges.

Question: Is a graded lattice with the above rank numbers and $n2^{n-1}$ edges, equivalent to $B_n$?

If it is not true, it would be useful (to me) to know up to which $n$ it is true.

Answer 1

True for $n \le 4$. For $n \le 3$ it is straightforward.

For $n=4$, inspect all 16-element vertically indecomposable graded lattices (listed here). The list has 1900132 lattices; of them, 103411 have rank sequence 1,4,6,4,1, and 249 have exactly 32 edges. But only 1 lattice has both, that's the $B_4$.