A refinement of the toral rank conjecture for 2-step nilpotent Lie algebras

It is known that the total (co)-homoloy of a 2-step nilpotent Lie algebra $\mathfrak{g}$ is at least $2^{\vert\mathfrak{z}\vert}$, where $\mathfrak{z}$is the center of $\mathfrak{g}$. We improve this result by showing that a better lower bound is $2^t$, where $t={\vert\mathfrak{z}\vert+\left[\frac{\vert v\vert+1}2\right]}$ and $v$ is a complement of $\mathfrak{z}$ in $\mathfrak{g}$. Furthermore, we provide evidence that this is the best possible bound of the form $2^t$.