Oguiso's rational-curve conjecture for Calabi–Yau threefolds

Let $X$ be a Calabi–Yau threefold. A Cartier divisor with nef-boundary class is a non-trivial Cartier divisor $D$ whose first Chern class lies in the boundary of the nef cone of $X$.

Oguiso's conjecture. If $X$ has a Cartier divisor $D$ with nef-boundary class, then $X$ contains a rational curve.

This is a weakened version of the general question whether Calabi–Yau threefolds contain rational curves. The assumption is strong: it requires $h^2(X)>1$ and a nonzero rational point on the boundary of the nef cone. The source gives no resolution status.