The radical splitting conjecture for connected algebraic groups

Let $G$ be a connected algebraic group over a field $k$. Radical splitting conjecture. One has

$$\operatorname{rd}_k(G) \leqslant 1.$$

This would follow from a positive answer to Tits's question of whether torsors under simple groups of type $E_8$ can be split by radicals; the claim concerns the resolvent degree of connected algebraic groups over arbitrary fields.