Combinatorial threshold-linear networks (CTLNs) are a special class of neural networks whose dynamics are tightly controlled by an underlying directed graph. In prior work, we showed that target-free cliques of the graph correspond to stable fixed points of the dynamics, and we conjectured that these are the only stable fixed points allowed. In this paper we prove that the conjecture holds in a variety of special cases, including for graphs with very strong inhibition and graphs of size $n \leq 4$. We also provide further evidence for the conjecture by showing that sparse graphs and graphs that are nearly cliques can never support stable fixed points. Finally, we translate some results from extremal combinatorics to upper bound the number of stable fixed points of CTLNs in cases where the conjecture holds.