2 problems

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.

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.