Characteristic-two conjecture for non-descending-hereditary reductive pairs

Let $(G,H)$ be a reductive pair of algebraic groups in characteristic $p$. The subgroup $H$ is $G$-descending hereditary (or $G$-dh) if, for every connected subgroup $K$ of $H$, $G$-complete reducibility of $K$ implies $H$-complete reducibility of $K$.

Characteristic-two conjecture. If $H$ is non-$G$-dh, then $p=2$.

This conjecture proposes that failure of descending hereditary behaviour for a reductive pair can occur only in characteristic two. The source presents it as a speculation based on empirical evidence, and does not state a resolution.