Let $P$ be a convex simplicial polytope in $\mathbb{R}^n$. Can we find a convex simplicial polytope $P_0$ in $\mathbb{R}^n$ combinatorially equivalent to $P$, satisfying the following condition: The vertices of $P_0$ are lattice points and for every facet $F$ of $P_0$ its vertices $v_1,\dots,v_n$ span $\mathbb{Z}^n$?

In other words, is a simplicial polytope combinatorially equivalent to a simplicial polytope such that the vertices of the facets and the origin form a unimodular simplex?