The integer decomposition property for nonempty $B_n(\Pi)$

Let $\mathfrak{S}_3$ denote the symmetric group on three letters, let $\Pi\subseteq\mathfrak{S}_3$, and let $B_n(\Pi)$ be the associated polytope. A conjecture on the integer decomposition property. If $B_n(\Pi)$ is nonempty, then $B_n(\Pi)$ has the integer decomposition property (IDP).