A gap theorem for positive Einstein metrics on the four-sphere

We show that there exists a universal positive constant $\varepsilon_0 > 0$ with the following property: Let $g$ be a positive Einstein metric on $S^4$. If the Yamabe constant of the conformal class $[g]$ satisfies $$Y(S^4, [g]) >\frac{1}{\sqrt{3}} Y(S^4, [g_{\mathbb S}]) - \varepsilon_0$$ where $g_{\mathbb S}$ denotes the standard round metric on $S^4$, then, up to rescaling, $g$ is isometric to $g_{\mathbb S}$. This is an extension of Gursky's gap theorem for positive Einstein metrics on the four-sphere.