In https://mathoverflow.net/q/126795, @Dietrich Burde says that He had read in the literature about rigid Lie algebras that no filiform Lie algebra can be rigid in $\mathcal{L}_n(\mathbb{C})$ (He could not verify such affirmation, and did not find a valid proof).
Someone knows which is (are) the paper(s) where I can read such affirmation? Is there any significant advance on this interesting subject over the last ten years?
The literature I have mentioned in my MO answer is the following:
J. M. Ancochea-Bermudez, J. R. Gómez-Martin, G. Valeiras, M. Goze: Sur les composantes irreductibles de la variete des lois d'algebres de Lie nilpotentes. J. Pure Appl. Algebra 106, No.1 (1996), 11-22.
In this article it is claimed without proof that no filiform Lie algebra in $\mathcal{L}_n(\mathbb{C})$ can be rigid.
Rigid Lie algebras are still studied intensively (by myself, and many other people like Paulo Tirao and his student Joan Felipe Herrera-Granada). You will find many articles if you search on rigid Lie algebras.