NP-hardness of deciding Nash equilibrium existence in single-peaked Schelling games

Let $\Lambda\in(0,1)$ be a peak. A Single-Peaked Jump Schelling Game and a Single-Peaked Swap Schelling Game are instances of the respective models, and $\operatorname{NE}$ denotes a Nash equilibrium. Equilibrium-existence complexity conjecture. For both the Single-Peaked Jump Schelling Game and the Single-Peaked Swap Schelling Game, it is NP-hard to decide whether a given instance admits an $\operatorname{NE}$. This is a main open problem for both game variants; the paper establishes hardness for finding equilibria via improving-response dynamics and notes NP-hardness of equilibrium-existence decisions when stubborn agents are allowed, but the stated result without that assumption remains open.