Stable constant mean curvature disks with circular boundary in homogeneous three-manifolds

Let $H>0$ and $κ,τ\in\mathbb{R}$ satisfy $4H^2+κ>0$ and $κ\leq 4τ^2$. We prove that an immersed stable disk of constant mean curvature $H$ and circular boundary in the space $\mathbb{E}(κ,τ)$ is a spherical cap of the canonical rotational $H$-sphere.