The nef anticanonical bundle converse to the Weak Lang Conjecture

Let $X$ be a smooth projective variety defined over a number field $K$. The anticanonical bundle $-K_X$ is nef when it has nonnegative degree on every curve $C\subset X$.

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

The first test case is provided by K3 surfaces. The assertion is known for curves and for many K3 surfaces, including a large class treated in the paper, but remains open for general K3 surfaces over number fields.