I have started studying regular subdivisions of the $n$-cube, and came across the following post: Regularity of Delaunay triangulation of a hypercube.

My question is whether the "standard triangulation" of the $n$-cube described in the answer of that post is always regular. I know that the answer is yes in two and three dimensions, and, intuitively, I would guess that it is also true for arbitrary $n$. I'd be grateful for any help.

(A subdivision is regular if it can be obtained by lifting the vertices to $\mathbb{R}^{n+1}$, and projecting the lower faces of the convex hull of the lifted vertices to $\mathbb{R}^n$.)