On the existence problem for tilted unduloids in $\mathbb{H}^2\times\mathbb{R}$

We study the existence problem for tilted unduloids in $\mathbb{H}^2\times\mathbb{R}$. These are singly periodic annuli with constant mean curvature $H>1/2$ in $\mathbb{H}^2\times\mathbb{R}$, and the periodicity of these surfaces is with respect to a discrete group of translations along a geodesic that is neither vertical nor horizontal in the Riemannian product $\mathbb{H}^2\times\mathbb{R}$. Via the Daniel correspondence we are able to reduce this existence problem to a uniqueness problem in the Berger spheres: if a pair of linked horizontal geodesics bounds exactly two embedded minimal annuli (for a fixed orientation of the boundary curves) then tilted unduloids in $\mathbb{H}^2\times\mathbb{R}$ exist.