The invariant-range conjecture for Blaschke-Minkowski homomorphisms

Let $\mathcal{K}^n$ be the space of convex bodies in $\mathbb{R}^n$. A Blaschke-Minkowski homomorphism is a map from convex bodies, equipped with Blaschke addition, to convex bodies, equipped with Minkowski addition, having the continuity, rotation-intertwining, and additivity properties considered in the paper. Its range is invariant under every Minkowski endomorphism if applying any Minkowski endomorphism to an element of the range produces another element of the range. Invariant-range conjecture. There exists an injective Blaschke-Minkowski homomorphism whose range is invariant under every Minkowski endomorphism. The paper states that this conjecture is equivalent to the conjecture that every Minkowski endomorphism is weakly monotone; consequently, it is part of the same open classification problem for Minkowski endomorphisms.