Solvable ideal subset conjecture for centers of Lie algebras

Let $\mathfrak{g}$ be a finite-dimensional minimal-graph-admissible Lie algebra associated with a minimal graph $G(V,E)$, and let $\mathcal{Z}(\mathfrak{g})$ denote its center. Solvable ideal subset conjecture. If $\mathcal{Z}(\mathfrak{g})$ is non-zero, then there exists a proper non-empty subset $W\subsetneq V$ that satisfies the ideal-graph-property and spans a solvable ideal of $\mathfrak{g}$. The source notes that this holds when $\dim(\mathfrak{g})=3$, while the general case remains unresolved.