  Given any $n \geq 2$, we show that if $\Omega \subsetneq \mathbb{R}^n$ is an
open convex domain (e.g. a half-space), and $u : \Omega \to \mathbb{R}$ is a
solution to the minimal surface equation which agrees with a linear function on
$\partial \Omega$, then $u$ must itself be linear.
