Asymptotic stability conjecture for the equilibria of the fractional-order dark-sector system

Let $p>0$ and let $X_{1,2}^*$ denote the two equilibria of the fractional-order system, given by

$$X_{1,2}^*=(\mp\frac{\sqrt{2}}{2}\sqrt{P+p},\pm\frac{\sqrt{2}}{4}(p-P)\sqrt{P+p},\frac{1}{2}(p+P))$$

where $P=\sqrt{p^2+4}$. Let $q$ be the fractional order. Asymptotic stability conjecture. The equilibria $X_{1,2}^*$ are asymptotically stable for $q\in(0,1)$ and all $p>0$. This extends the proved stability result for $p=5$; the assertion is supported by computational evidence, while the general case is not established analytically.