  Given a smooth projective variety $X$ over a number field $k$ and $P\in
X(k)$, the first author conjectured that in a precise sense, any sequence that
approximates $P$ sufficiently well must lie on a rational curve. We prove this
conjecture for smooth split toric surfaces conditional on Vojta's conjecture.
More generally, we show that if $X$ is a $\mathbb{Q}$-factorial terminal split
toric variety of arbitrary dimension, then $P$ is better approximated by points
on a rational curve than by any Zariski dense sequence.
