Approximating rational points on surfaces

Let X be a smooth projective algebraic variety over a number field k and P in X (k). In 2007, the second author conjectured that, in a precise sense, if rational points on X are dense enough, then the best rational approximations to P must lie on a curve. We present a strategy for deducing a slightly weaker conjecture from Vojta's conjecture, and execute this strategy for the full conjecture for split surfaces.