McKinnon's rational-curve conjecture for best approximation constants

Let $X$ be a variety over a number field $K$ and let $L$ be an ample line bundle. Suppose that $Q\in X(K)$ and that there exists a rational curve defined over $K$ passing through $Q$ on $X$.

McKinnon's conjecture. There exists a rational curve $C$ on $X$ passing through $Q$ achieving the best approximation constant at $Q$ with respect to $L$.

This conjecture formulates a local analogue of the relationship between rational points and rational curves suggested by the Batyrev–Manin heuristic. It asserts that, whenever a rational curve through the point exists, the optimal Diophantine approximation to that point is achieved on some such curve.