Integer decomposition property of free sums of convex polytopes

Let $\mathcal{P} \subset \mathbb{R}^{d}$ and $\mathcal{Q} \subset \mathbb{R}^e$ be integral convex polytopes of dimension $d$ and $e$ which contain the origin of $\mathbb{R}^{d}$ and $\mathbb{R}^e$, respectively. In the present paper, under some assumptions, the necessary and sufficient condition for the free sum of $\mathcal{P}$ and $\mathcal{Q}$ to possess the integer decomposition property will be presented.