The target-free clique conjecture for combinatorial threshold-linear networks

Let $G$ be a directed graph on $n$ nodes, and let $W=W(G;\varepsilon,\delta)$ be the associated combinatorial threshold-linear network (CTLN). A subset $\sigma\subseteq[n]$ is a clique if $G|_{\sigma}$ has all bidirectional edges between distinct vertices. A vertex $v\notin\sigma$ is a target of $\sigma$ if every $j\in\sigma$ has $j\to v$ in $G$, and $\sigma$ is target-free if it has no external target. Assume

$$\delta>0,\qquad 0<\varepsilon<\frac{\delta}{\delta+1}.$$

The target-free clique conjecture. A subset $\sigma$ is the support of a stable fixed point if and only if $\sigma$ is a target-free clique. Target-free cliques are known to support stable fixed points, and the converse has been proved in several special cases, including oriented and symmetric graphs, networks with at most four nodes, and certain parameter ranges; the general statement remains open.