The disklike characterization conjecture for maximal compatible transfer systems

Let $G$ be a group and let $\mathcal{O}$ be a disklike $G$-transfer system. For transfers $r,e\in\mathcal{O}$, write $r<e$ when $e$ restricts into $r$ in the poset of restrictions, and write $r<^S e$ and $r<^F e$ when this restriction is respectively a compatibility success or failure. Let $\mathrm{M}(\mathcal{O})$ denote the maximal compatible transfer system associated to $\mathcal{O}$, and let

$$\mathrm{M}(\mathcal{O})^c\coloneqq\mathcal{O}\backslash\mathrm{M}(\mathcal{O}).$$

Disklike characterization conjecture. The maximal compatible transfer system is

$$\mathrm{M}(\mathcal{O})=\{e\in\mathcal{O}\mid\forall r<e,\ r<^S e\},$$

where every restriction is in $\mathcal{O}$, and its complement is

$$\mathrm{M}(\mathcal{O})^c=\{e\in\mathcal{O}\mid\exists r<e,\ r<^F e\}.$$

This conjecture would give a direct characterization of maximal compatibility for arbitrary disklike transfer systems by checking the compatibility successes and failures of their restrictions. The surrounding results establish related structural and computational properties, while the proposed characterization is presented as a conjecture and its general validity remains open.