Vergne's conjecture is still open. It says that there is no complex $n$-dimensional nilpotent Lie algebra which is rigid in the variety $\mathcal{L}_n(\mathbb{C})$ of all $n$-dimensional complex Lie algebras. The Grunewald and O’Halloran conjecture, which implies Vergne's conjecture, states that every complex nilpotent Lie algebra is the proper degeneration of another Lie algebra of the same dimension.
The Heisenberg Lie algebra, with $[x,y]=z$ is rigid in $\mathcal{N}_3(\mathbb{C})$, but
not in $\mathcal{L}_3(\mathbb{C})$, since it is a proper degeneration of $sl_2(\mathbb{C})$.

R. Carles gives a necessary condition for a nilpotent Lie algebra $L$ to be rigid in $\mathcal{L}_n(\mathbb{C})$: it must be characteristically nilpotent, such that every ideal
of codimension $1$ in $L$ is again characteristically nilpotent. It is not clear, whether
Carles condition is also sufficient. Certainly the condition to be characteristically nilpotent is not sufficient. It is easy to construct characteristically nilpotent Lie algebras, which are not rigid. But even for the stronger condition of Carles, I found
filiform nilpotent Lie algebras of dimension $n\ge 13$, which are characteristically nilpotent, and every ideal of codimension $1$ also being characteristically nilpotent.
Unfortunately I do not know whether these algebras are rigid or not
(there is a claim in the literature that no filiform nilpotent Lie algebra can be rigid in $\mathcal{L}_n(\mathbb{C})$, but I could not verify this, and did not find a valid proof).