One-dimensional subgroup tessellation conjecture

Let $G$ be a Lie group, let $H$ be a closed subgroup, and let $d(H)$ denote the invariant used in the paper to measure the dimension relevant to tessellations of $G/H$. One-dimensional subgroup conjecture. If

$$d(H)=1,$$

then $G/H$ does not have a tessellation. The source says that this conjecture may be implicit in work of Oh and Witte, but gives no resolution status.