Gelfand-Tsetlin polytopes and the integer decomposition property

Let $P$ be the Gelfand--Tsetlin polytope defined by the skew shape $\lambda/\mu$ and weight $w$. In the case corresponding to a standard Young tableau, we completely characterize for which shapes $\lambda/\mu$ the polytope $P$ is integral. Furthermore, we show that $P$ is a compressed polytope whenever it is integral and corresponds to a standard Young tableau. We conjecture that a similar property hold for arbitrary $w$, namely that $P$ has the integer decomposition property whenever it is integral. Finally, a natural partial ordering on GT-polytopes is introduced that provides information about integrality and the integer decomposition property, which implies the conjecture for certain shapes.