Gizatullin's automorphism conjecture for complements of smooth cubic surfaces

Let $S$ be a smooth cubic surface in $\mathbb P^3$. Gizatullin's automorphism conjecture. Every automorphism of the affine variety $\mathbb P^3\setminus S$ is induced by an automorphism of $\mathbb P^3$; equivalently,

$$\operatorname{Aut}(\mathbb P^3\setminus S)=\operatorname{Aut}(\mathbb P^3,S).$$

This is a folklore conjecture concerning automorphisms of affine complements; the source later relates it to results on cylindrical complements, but does not state a resolution here.