Polyhedrality conjecture for the k-halfspace closure

Let $k\in\mathbb{N}$ be fixed. For a rational polyhedron $P$, define the $k$-halfspace closure by

$$\mathcal{H}_k(P):=\bigcap_{\substack{k\text{-halfspace rational }\\\\ \text{relaxation }Q\text{ such that }\\\P\subseteq Q}} Q_I,$$

where $Q_I$ denotes the integer hull of $Q$. Polyhedrality conjecture for the $k$-halfspace closure. For any fixed natural number $k\in\mathbb{N}$, and any rational polyhedron $P$, $\mathcal{H}_k(P)$ is a rational polyhedron. This asks whether the closure obtained by intersecting integer hulls of all $k$-halfspace rational relaxations remains finitely describable by rational linear inequalities; the source gives no resolution, so the conjecture remains open.