Uniqueness conjecture for maximal solvable extensions of nilpotent Lie algebras

Let ${\mathfrak n}$ be a complex nilpotent Lie algebra that is not characteristically nilpotent. Let ${\mathfrak s}$ and $\tilde{{\mathfrak s}}$ be solvable Lie algebras with nilradical ${\mathfrak n}$, of maximal dimension in the sense that no such solvable algebra of larger dimension exists. Uniqueness conjecture. The Lie algebras ${\mathfrak s}$ and $\tilde{{\mathfrak s}}$ are isomorphic.

This conjecture is motivated by the observation that, in the classes of nilradicals investigated in the cited literature, maximal solvable extensions over the complex numbers are unique up to isomorphism. Whether this uniqueness holds for every complex nilpotent Lie algebra that is not characteristically nilpotent remains open.