The connection-blocking conjecture for homogeneous spaces

Let $G$ be a connected Lie group and let $\Gamma \subset G$ be a lattice. Set $M=G/\Gamma$. The connection curves of $M$ are the orbits of one-parameter subgroups of $G$. The space $M$ is blockable if every pair of points can be blocked by a finite set disjoint from the pair.

Connection-blocking conjecture. The space $M$ is blockable if and only if

$$G=\mathds{R}^n,$$

i.e., if and only if $M$ is a torus.

This is the homogeneous-space counterpart of the security conjecture for closed Riemannian manifolds. The paper proves that all quotients of $Sol$ are non-blockable, but the stated characterization for general connected Lie groups with lattices remains open.