By lattice points, I will always mean points in $\mathbb{Z}^n$ and all polytopes here are convex rational polytopes.

If $P$ is an integral polytope, the counting function for the number of lattice points inside $nP$ is a polynomial $p(n)$. This is the Erhart polynomial, and if $P$ does not have integral vertices, $p(n)$ is in general just a quasi-polynomial.

A integrally closed polytope is a polytope, such that each lattice point in $nP$ is a sum of exactly $n$ lattice points in $P$.

To the question: Now, let $P$ be a non-integral polytope, but its Erhart quasi-polynomial is in fact a polynomial. Let $P'$ be the convex hull of the lattice points in $P'$.

Are there examples $P$ for which $P'$ is not integrally closed?

Motivation: Being integrally closed is a quite rare property for a polytope (in high dimensions). Having a polynomial Erhart function, when only a quasi-polynomial is expected is also rare. Therefore, the chance that these coincide is even rarer, so one would expect a lot of examples above, or these properties are related somehow.