The strong Tits Centre Conjecture for vector edifices

Let $V_G({\mathbb K})$ be the vector edifice associated with $G$. A cone is a subset stable under multiplication by ${\mathbb K}^+$. A closed convex cone $C$ is completely reducible if every $x\in C$ has an opposite in $C$. Let $\Gamma$ be a subgroup of $\operatorname{Aut}(V_G({\mathbb K}))$ stabilising $C$.

Strong Tits Centre Conjecture for vector edifices. If $C$ is a closed convex non-completely-reducible cone in $V_G({\mathbb K})$, then $C$ has an unopposed $\Gamma$-centre.

This is the vector-edifice formulation of the strong Tits Centre Conjecture. The supplied text gives no evidence that this formulation has been resolved.