The unipotency conjecture for twisted function algebras

Let $G$ be a pro-algebraic group, let ${\cal O}(G)$ be its coordinate Hopf algebra over $k$, and let $J$ be a Hopf $2$-cocycle for ${\cal O}(G)$. Write ${\cal O}(G)^J$ for the Hopf algebra obtained by twisting ${\cal O}(G)$ by $J$, and let $S$ denote its antipode.

Unipotency conjecture. The operator

$$S^2$$

is unipotent on ${\cal O}(G)^J$.

This predicts that twisting the function algebra of any pro-algebraic group by a Hopf $2$-cocycle cannot produce non-unipotent behavior for the square of the antipode. The supplied text gives no resolution, so the conjecture remains open.