Uniform boundedness conjecture for semisimple algebraic groups

Let $G$ be a semisimple linear algebraic group over an infinite field $k$. Write $G^+(k)$ for the subgroup generated by the subgroups $R_u(Q)(k)$, where $Q$ ranges over all minimal $k$-parabolic subgroups of $G$. Uniform boundedness conjecture. Then $G^+(k)$ is uniformly bounded. This conjecture concerns the boundedness behaviour of semisimple algebraic groups; the paper proves that boundedness of $G^+(k)$ implies uniform boundedness and establishes explicit bounds under additional hypotheses, but the general assertion remains open.