Polyhedrality conjecture for the k-dimensional projection closure

Let $k\in\mathbb{N}$ be fixed. For a rational polyhedron $P\subseteq\mathbb{R}^n$, define the $k$-dimensional projection closure by

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

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 $L$. Polyhedrality conjecture for the $k$-dimensional projection closure. For any fixed natural number $k\in\mathbb{N}$, and any rational polyhedron $P$, $\mathcal{P}_k(P)$ is a rational polyhedron. This asks whether the intersection over all $k$-dimensional projections is a rational polyhedron; the source poses the question without resolving it.