The stable fixed point conjecture for target-free cliques in combinatorial threshold-linear networks

Let $G$ be a directed graph on vertex set $[n]=\{1,\ldots,n\}$, and let $\sigma\subseteq[n]$. A target-free clique is a clique in $G$ with no common target outside the clique. The subset $\sigma$ is the support of a stable fixed point of the combinatorial threshold-linear network associated to $G$ if and only if it is a target-free clique.

Stable fixed point conjecture. A subset $\sigma\subseteq[n]$ is the support of a stable fixed point if and only if $\sigma$ is a target-free clique.

This conjecture extends the stated fact that target-free cliques are exactly the supports of fixed points, and that these fixed points are stable. It also includes the singleton case, where target-free cliques are precisely sinks. The surrounding discussion presents this as a conjecture based on the observed graphs; its general status is not established in the supplied text.