The automorphism conjecture for complements of del Pezzo surfaces

Let $S$ be the del Pezzo surface under consideration, and let $\mathbb P$ denote the ambient projective threefold in which it is embedded. The automorphism conjecture for del Pezzo complements. The surface $S$ contains no $(-K_S)$-polar cylinder if and only if

$$\operatorname{Aut}(\mathbb P\setminus S)=\operatorname{Aut}(\mathbb P,S).$$

The conjecture generalizes the smooth cubic-surface case and is motivated by results linking polar cylinders to unipotent automorphisms; its general validity remains open.