Caprace–Marquis conjecture on split spherical BN-pairs of anisotropic groups

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 trivial when $B=G(k)$, and virtually trivial when its associated building has the corresponding virtual triviality.

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

This conjecture is presented as a converse to the Borel–Tits result that isotropic reductive groups have canonical non-trivial split spherical $BN$-pairs. The paper proves the assertion in several situations, but the general conjecture remains open.