  Let $X$ be a smooth projective split horospherical variety over a number
field $k$ and $x\in X(k)$. Contingent on Vojta's conjecture, we construct a
curve $C$ through $x$ such that (in a precise sense) rational points on $C$
approximate $x$ better than any Zariski dense sequence of rational points. This
proves a weakening of a conjecture of McKinnon in the horospherical case. Our
results make use of the minimal model program and apply as well to
$\mathbb{Q}$-factorial horospherical varieties with terminal singularities.
