Since ${\rm Spec}({K^{\circ}})$ is affine, ampleness is the same as relatively ample. It remains to check that the restriction of $\omega+{\varepsilon}\theta$ to the special fibre is ample (see [Gro66, 9.6.4 and 9.6.5]). The ample cone on the special fiber is the interior of the nef cone. This proves immediately the claim. ∎