Fine-grained magma-graph representation conjecture for Lie algebras

Let $\mathfrak{g}$ be a complex finite-dimensional Lie algebra, and let $(\mathcal{M},\delta)$ be an abelian magma such that $\mathfrak{g}$ is $\mathcal{M}$-magma-graded, with the $\mathcal{M}$-gradation having the finest granularity. Let $G_{\mathfrak{g}}(V,E)$ be the graph associated with this pair by the modified graph-construction algorithm. Fine-grained magma-graph conjecture. The graph faithfully represents the internal structure of $\mathfrak{g}$: $\mathfrak{g}$ is non-solvable if and only if the graph contains a self-contained subgraph induced by a closed directed walk; the derived-series and lower-central-series graph sequences yield the corresponding Lie algebras; ideal-graph-property subsets span ideals; simplicity implies a closed directed walk inducing the whole vertex set; and semisimplicity implies that every vertex lies on a closed directed walk inducing a self-contained subgraph. The source presents this as a stronger version of an earlier conjecture and leaves it open.