The weak monotonicity conjecture for Minkowski endomorphisms

Let $n$ be the dimension, and let $\mathcal{K}^n$ denote the space of convex bodies in $\mathbb{R}^n$. A Minkowski endomorphism is a continuous, rotation-intertwining, Minkowski-additive map $\Phi:\mathcal{K}^n\to\mathcal{K}^n$; it is weakly monotone if it is monotone with respect to set inclusion on convex bodies whose Steiner point is the origin. Weak monotonicity conjecture. For $n\geq 3$, every Minkowski endomorphism is weakly monotone. The conjecture would give a classification of all Minkowski endomorphisms through the representation by weakly positive measures currently known under the weak-monotonicity assumption; it is known in dimension $2$ but remains open in dimensions at least $3$.