Nikolayevsky–Nikonorov's characterization conjecture for Ricci-negative solvable Lie algebras

Let $\mathfrak{s}$ be a solvable Lie algebra with nilradical $\mathfrak{n}$. For a nilpotent Lie algebra $\mathfrak{n}$, let $\mathfrak{t}(\mathfrak{n})$ be a maximal torus of diagonalizable derivations, and write $\operatorname{ad}(Y)|_{\mathfrak{n}}^{\mathbb{R}}$ for the real semisimple part of the restriction of $\operatorname{ad}(Y)$ to $\mathfrak{n}$. Nikolayevsky–Nikonorov's characterization conjecture. For each nilpotent Lie algebra $\mathfrak{n}$, there is an open and convex cone $\mathfrak{c}\subset\mathfrak{t}(\mathfrak{n})$ such that $\mathfrak{s}$ admits a $\operatorname{Ric}<0$ metric if and only if there exists $Y\in\mathfrak{s}$ such that $\operatorname{ad}(Y)|_{\mathfrak{n}}^{\mathbb{R}}\in\mathfrak{c}$, up to automorphism conjugation. This would provide a complete characterization of solvable Lie algebras admitting Ricci-negative metrics; the rank-one reduction is known in the Einstein case, but the Ricci-negative characterization remains unresolved.