Converse to Borel–Tits for anisotropic reductive groups

Let $G$ be a reductive algebraic $k$-group that is anisotropic over $k$, meaning that it has no proper $k$-parabolic subgroup. A split spherical BN-pair for $G(k)$ is a pair of subgroups $(B,N)$ forming a spherical BN-pair whose associated root groups split in the relevant sense.

Converse to Borel–Tits. Every split spherical BN-pair for $G(k)$ is trivial.

The canonical BN-pair supplied by the Borel–Tits theorem is trivial in the anisotropic case, and this conjecture asks whether all split spherical BN-pairs on the group of rational points must likewise be trivial. The paper establishes this when $k$ is a perfect field or a local field, while the statement for arbitrary fields remains open.