I am looking for a reference stating that

If a graph $z=f(x,y)$ over a convex domain $D$ is minimal, then it is area-minimizing.

• 5.4.18 in Federer's "Geometric measure theory" and Lemma 1.1. in Colding--Minicozzi's "A Course in Minimal Surfaces" state that it is true with respect to $D\times \mathbb R$.

• 6.1 in Morgan's "Geometric measure theory" states it right, but (at least formally) the proof only shows that it is area-minimizing among oriented surfaces.