Are all marked order polytopes normal?

Question

Richard Stanley showed that order polytopes have a unimoudlar triangulation. In particular, this implies that they are integrally closed/normal.

One can generalize order polytopes to marked order polytopes, where the smallest and largest elements are not always 0 and 1 respectively, but other integers, and the poset do not have unique top and bottom.

For example, Gelfand-Tsetlin polytopes can be realized as the construction above, and it is quite easy to show that these particular polytopes are integrally closed.

However, is it known if ALL marked order polytopes are integrally closed?

Reminder: A convex polytope $P$ with integer vertices is integrally closed if we have that each integer point $p$ in $kP$ can be expressed as $p=p_1+\dots+p_k$ where each $p_i$ is an integer point in $P$.

Answer 1

I have found a positive answer to this question: One can interpret $p$ as a point in some $kP'$ where $P'$ is the poset where the unknowns are totally ordered, and have fixed lower and upper bounds. This poset $P'$ can then be seen as a face in a GT-polytope, so using that the GT-polytopes are integrally closed, it follows that $p$ can be expressed as a sum of points in $P'$, and the inequalities between coordinates are by construction compatible with those in $P$.

Actually, one can show a stronger statement, that each such polytope has regular, unimodular triangulation.