Simply connected smooth projective surfaces with thin integral points

Establish whether there exists, over the rational numbers, a smooth projective surface analogous to the simply connected affine cubic surface whose Zariski-dense set of integral points is thin.

Background

The paper constructs a simply connected smooth affine cubic surface over Q whose integral points over Z are Zariski-dense but thin, thereby giving a negative answer to the Hilbert Property question for integral points on simply connected varieties over Q.

The authors explain that the corresponding Hilbert Property is expected to hold after a suitable finite extension of the ground field. They then isolate a further unresolved geometric question: whether a similar counterexample can occur for a smooth projective surface defined over Q.