Inequivalence conjecture for higher-dimensional halfspace and projection closures

Let $k\geq 3$ and $n\geq k$. For a rational polyhedron $P\subseteq\mathbb{R}^n$, let $\mathcal{H}_k(P)$ be its $k$-halfspace closure, and let

$$\mathcal{P}_k(P):=\bigcap_{L\in\mathcal{L}_k}\bigl(P_{I,L}+L^\perp\bigr),$$

where $\mathcal{L}_k$ is the family of $k$-dimensional subspaces, $P_{I,L}$ is the integer hull in the corresponding projection, and $L^\perp$ is the orthogonal complement of $L$. Inequivalence conjecture. For any fixed natural number $k\geq 3$, and any $n\geq k$, there are instances of rational polyhedra $P\subseteq\mathbb{R}^n$ such that

$$\mathcal{H}_k(P)\neq\mathcal{P}_k(P).$$

The claim is motivated by the failure in dimensions three and higher of representing integer hulls as intersections of corner polyhedra; the source notes that it is true when $n=k$, but gives no general resolution for the stated range.