Wiegold's conjecture for nilpotent Lie algebras

Let $\mathfrak{g}$ be a nilpotent Lie algebra over a field $\mathbb{F}$. For an element $x\in\mathfrak{g}$, define its breadth $b(x)$ by

$$\dim\mathfrak{g}-\dim C_{\mathfrak{g}}(x)=b(x),$$

where $C_{\mathfrak{g}}(x)$ is the Lie centralizer of $x$ in $\mathfrak{g}$. Wiegold's conjecture for nilpotent Lie algebras. If $\dim\mathfrak{g}'>n(n-1)/2$ for some non-negative integer $n$, then $\mathfrak{g}$ can be generated by elements of breadth at least $n$. This is the Lie algebra analogue of Wiegold's finite-group conjecture, and the paper's stated goal is to prove it.