Behrend's vanishing conjecture for canonical parabolic subgroups

Let $C$ be a smooth projective curve and let $G$ be a reductive group scheme over $C$. Let $P\subset G$ be the canonical parabolic subgroup. Behrend's conjecture. One has

$$H^0(C,\operatorname{Lie}(G)/\operatorname{Lie}(P))=0.$$

This is the group-scheme formulation of Behrend's cohomological conjecture, which is connected to the rationality of canonical destabilizing reductions and to the construction of moduli spaces of semistable bundles. The paper proves the assertion in broad cases but exhibits failure for $G_2$ in characteristic $2$, so it is refuted without additional hypotheses.