Corvaja–Zannier's conjecture on the Hilbert Property

Let $K$ be a number field and let $X$ be a variety over $K$. Assume that $X(K)$ is Zariski dense and that $X$ has a smooth, projective model over $K$ which is algebraically simply connected. Corvaja–Zannier's conjecture. Then $X$ has the Hilbert Property. The conjecture asserts that, for smooth projective simply connected varieties, Zariski density of rational points is the only obstruction to the Hilbert Property. It is known in particular for Kummer surfaces associated to products of two elliptic curves, but remains open in general.