Vergne's conjecture on rigid complex nilpotent Lie algebras

Let $\mathrm{L}_n$ be the algebraic variety of complex Lie algebras of dimension $n$. A Lie algebra in $\mathrm{L}_n$ is rigid if its isomorphism class is open in the relevant algebraic-geometric topology on $\mathrm{L}_n$.

Vergne's conjecture. There are no rigid complex nilpotent Lie algebras in $\mathrm{L}_n$.

The source presents this as a well-known conjecture in Lie theory and gives no resolution or partial result for it in the supplied context, so its status remains open here.