The converse to the Weak Lang Conjecture for varieties with negative canonical bundle

Let $X$ be a smooth projective variety defined over a number field $K$. Its canonical bundle is negative when $-K_X$ is ample.

The negative-canonical-bundle converse. If the canonical bundle $K_X$ of $X$ is negative, then for some finite extension $K'$ of $K$ the set $X(K')$ of $K'$-rational points of $X$ is Zariski dense.

The conjecture is true for curves and surfaces, where the hypothesis ensures that $X$ is rational; the first real test cases are Fano threefolds. The paper proves the assertion for smooth quartic hypersurfaces in projective space, but the general statement remains open.