Classification conjecture for smooth affine cubic surfaces with infinite discrete automorphism group

Let $S$ be a smooth, affine, cubic surface whose automorphism group is infinite and discrete. A classification conjecture asserts that $S$ is either a Markov-type surface, or the curve at infinity $\bar B$ is nodal and $\bar S$ is singular at the node. This would extend the known classification of smooth Del Pezzo compactifications with nodal boundary, for which the degree is at least $4$; the remaining cases for affine cubic surfaces are not all covered.