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.