Intermediate reduced degrees for the int* degree of a polytope

Let ${\mathcal P} \subset {\mathbb R}^n$ be a normal polytope of dimension $n$ with $({\mathcal P} \setminus \partial {\mathcal P}) \cap {\mathbb Z}^n \neq \emptyset$. For ${\mathbf a} \in N({\mathcal P} \setminus \partial {\mathcal P}) \cap {\mathbb Z}^n$, its reduced ${\mathcal P}$-degree is the smallest integer $r \geq 1$ for which there are ${\mathbf a}_0 \in r({\mathcal P} \setminus \partial {\mathcal P}) \cap {\mathbb Z}^n$ and ${\mathbf a}' \in (N-r){\mathcal P} \cap {\mathbb Z}^n$ such that ${\mathbf a}={\mathbf a}_0+{\mathbf a}'$. The int degree assertion.* If the int* degree of ${\mathcal P}$ is $d$, then for each $1 \leq i < d$, there is ${\mathbf a} \in N({\mathcal P} \setminus \partial {\mathcal P}) \cap {\mathbb Z}^n$ whose reduced ${\mathcal P}$-degree is $i$. The reduced ${\mathcal P}$-degree is known to be at most $n-1$, while the int* degree records the biggest such degree; the assertion asks whether every intermediate degree occurs whenever the maximum is $d$.