Two-component LSP compatibility conjecture for transfer systems

Let $G$ be a group and let $T$ be a $G$-transfer system whose underlying graph has two connected components, one containing $e$ and one containing $G$. Suppose the connected component containing $G$ has only the edge $G\to G$. Let LSP denote the property used in the source for transfer systems, and let the transfer system generated by adding the edge $e\to G$ to $T$ be the corresponding enlargement of $T$.

Two-component LSP compatibility conjecture. Then $T$ is not LSP and is compatible with the transfer system generated by adding the edge $e\to G$ to $T$.

This conjecture proposes a structural criterion for failure of the LSP property and a resulting compatibility relation. The supplied text presents it as a conjecture and indicates that the authors have progress toward proofs, but does not establish it in full.