Uniqueness and sharp existence bound for invariant CMC tubes

Let $c$ be a screw motion geodesic in a homogeneous space with parameters $c\kappa$ and $c\tau$, and let a tube invariant by translations along $c$ have constant mean curvature $H$. Sharp existence and uniqueness conjecture. There exists a unique constant $\mathcal{H}_0\geq0$ such that an invariant constant-mean-curvature tube along $c$ exists if and only if $H>\mathcal{H}_0$. Whenever such a tube exists, it is unique up to isometry. The conjecture is motivated by numerical evidence and extends uniqueness results known for $\operatorname{Nil}_3$, $\mathbb{S}^2\times\mathbb{R}$, and $\mathbb{S}^3_\mathrm{Berg}$ with horizontal geodesic $c$; the sharpness of the existence bound and uniqueness in the general case remain open.