The composite-graph permitted motif conjecture

Let $G$ be a composite graph with skeleton $\widehat G$. A permitted motif is a motif that supports a fixed point of the associated competitive threshold-linear network. Assume that $\widehat G$ is a permitted motif.

Composite-graph permitted motif conjecture. $G$ is a permitted motif if and only if every component $G_i$ is permitted.

This conjecture asks whether the characterization proved for disjoint, clique, and cyclic unions extends to all composite graphs whose skeletons are permitted motifs. The supplied text gives no resolution, so its status remains open.