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

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