McKinnon's rational-curve conjecture for best approximations

Let $V$ be an algebraic variety defined over a field $k$, let $D$ be an ample divisor on $V$, and let $P\in V(k)$ be a $k$-rational point. Assume that there is a rational curve $C$ on $V$, defined over $k$, with $P\in C(k)$. A best-approximation sequence to $P$ on $V$ with respect to $D$ is a sequence achieving the best possible approximation constant. McKinnon's rational-curve conjecture. Under these assumptions, a sequence of best approximation to $P$ on $V$ with respect to $D$ exists, and may be chosen to lie along some rational curve through $P$. The paper proves the conjecture for a wide range of examples and derives it for many more examples from Vojta's Main Conjecture, but notes that it remains open for general rational surfaces.