Kernel-intersection criterion for graph-admissible Lie algebras

Let $\mathfrak{g}$ be a finite-dimensional minimal-graph-admissible Lie algebra associated with a minimal graph $G(V,E)$, with $\dim(\mathfrak{g})\geq 2$, and let $\boldsymbol{\alpha}^{(\ell)}$ and $\mathcal{N}$ be the matrices and index set used in the source. Kernel-intersection conjecture. If

$$\bigcap_{\ell\in\mathcal{N}}\operatorname{Ker}\{\boldsymbol{\alpha}^{(\ell)}\}=\{0\},$$

then no proper non-empty subset $W\subsetneq V$ satisfies the ideal graph property and spans a non-solvable ideal of $\mathfrak{g}$. The conjecture is presented as a refinement related to the preceding center and solvable-ideal conjecture; its general validity is unresolved.