Caprace–Marquis weaker conjecture on virtually trivial split spherical BN-pairs

Let $k$ be a field and let $G$ be a reductive algebraic $k$-group that is anisotropic over $k$. A split spherical $BN$-pair $(B,N)$ of $G(k)$ is a saturated $BN$-pair such that $B=U\rtimes T$ with $U$ nilpotent. It is virtually trivial in the sense used by Caprace and Marquis.

Caprace–Marquis weaker conjecture. Every split spherical $BN$-pair of $G(k)$ is virtually trivial.

This is explicitly presented as a weaker form of the conjecture that every such $BN$-pair is trivial. The paper contributes results toward these statements, while the general weaker conjecture remains open.