The strong Tits Centre Conjecture for spherical edifices

Let $G$ be a connected reductive group over a field $k$, let ${\mathbb K}$ be the relevant ordered field, and let $\Delta_G({\mathbb K})$ be its spherical edifice. A closed convex subset $\Sigma$ is completely reducible if every $x\in\Sigma$ has an opposite in $\Sigma$. If $\Gamma\subseteq\operatorname{Aut}(\Delta_G({\mathbb K}))$ stabilises $\Sigma$, a point $x\in\Sigma$ is a $\Gamma$-centre when $x\neq0$ and the simplex $\sigma_x$ is fixed by $\Gamma$; it is unopposed when it has no opposite in $\Sigma$.

Strong Tits Centre Conjecture. If $\Sigma$ is a closed convex non-completely-reducible subset of $\Delta_G({\mathbb K})$, then $\Sigma$ has an unopposed $\Gamma$-centre.

This is a point-edifice analogue of the Tits Centre Conjecture and is weaker than the version asserting a $\Gamma$-fixed point, which the source notes is false without an isometric-action hypothesis. Its general status is not specified in the supplied text.