This is a sequel of this [post][1].  

The boolean lattice $B_n$ is [graded][2] 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.

  [1]: https://mathoverflow.net/q/248566/34538
  [2]: https://en.wikipedia.org/wiki/Graded_poset