Two-step nilpotent minimal-graph-admissibility conjecture

Let $\mathfrak{g}$ be a finite-dimensional $k$-step nilpotent Lie algebra. Nilpotent minimal-graph-admissibility conjecture. If $k\leq 2$, then $\mathfrak{g}$ is minimal-graph-admissible; if $k>2$, then $\mathfrak{g}$ is not necessarily minimal-graph-admissible, and there exist $k$-step nilpotent Lie algebras that fail to be minimal-graph-admissible. The motivation is that adjoint actions commute in the two-step case, whereas this need not hold for three-step nilpotent Lie algebras; the conjecture remains open.