Folklore conjecture on automorphisms of affine cubic hypersurfaces

Let $X$ be a smooth cubic hypersurface in the complex projective space~$\mathbb{P}^4$, and let $S_X$ be its hyperplane section. The complement $X\setminus S_X$ is an affine cubic hypersurface in~$\mathbb{A}^4$. Suppose that the cubic surface $S_X$ is smooth. Folklore conjecture. Then

$$\mathrm{Aut}\left(X\setminus S_X\right)=\mathrm{Aut}\left(X,S_X\right).$$

In particular, the group $\mathrm{Aut}(X\setminus S_X)$ is finite. This claim concerns the rigidity of automorphisms of affine cubic hypersurfaces obtained as complements of smooth hyperplane sections; the source presents it as folklore and does not provide resolution evidence.