Rational curves on Calabi–Yau varieties over finite fields

Let $X$ be a smooth projective variety over a finite field $k$. Assume that $X$ has trivial canonical class and that $X_{\bar{k}}$ has trivial algebraic fundamental group. An algebraic point means a point of $X(\bar{k})$. Rational-curve conjecture. Every algebraic point of $X$ lies on a rational curve $C\subset X$, defined over $\bar{k}$. The statement is motivated by examples involving Kummer surfaces and Calabi–Yau varieties, but its general validity is not established.