Uniqueness conjecture for hd-compactifications of real-reductive Lie groups

Let $G$ be a real reductive Lie group with compact center, and let an hd-compactification of $G$ be a compact manifold with corners $\overline{G}$ containing $G$ as its interior and satisfying inversion, b-normality, b-transitivity, and minimality. Equivalence is required to intertwine the right and left actions. Uniqueness conjecture. The hd-compactification of $G$ is unique up to equivalence, intertwining the right and left actions, and therefore defines a functor from real-reductive Lie groups with compact center and isomorphisms to compact manifolds with corners and diffeomorphisms. The claim asserts canonical functoriality of the compactification construction for this class of Lie groups; the supplied text gives no evidence resolving it.