Consider an $n$-simplex with vertices given by $(0,0,\dots,r_i,r_{i+1},\dots,r_n)$ where $r_1,\dots,r_n$ are given natural numbers, and $i=0,1,\dots,n+1$.
Does this simplex admit a regular, unimodular triangulation?
Consider an $n$-simplex with vertices given by $(0,0,\dots,r_i,r_{i+1},\dots,r_n)$ where $r_1,\dots,r_n$ are given natural numbers, and $i=0,1,\dots,n+1$.
Does this simplex admit a regular, unimodular triangulation?
The answer is yes. Consider the hyperplanes $x_i=x_j+m$ and $x_i=m$ for all $i$ $j$ and integers $m$. These hyperplanes triangulate the entire space into unimodular simplices, and the hyperplanes are compatible with the hyperplanes that determine the polytope in the question (meaning each face is determined by an intersection of some of these hyperplanes). This induces a triangulation of the polytope, and it should be clear that this is regular.
EDIT: This is not a correct answer to the question as stated, I had a different question in mind. I was wondering if Gelfand-Tsetlin polytopes admits unimodular triangulations (which they do), and the original question I had in mind (but did not write correctly) would have implied that.