Boundary-vertex conjecture for minimal covering bodies

Let $K$ be a convex body and let $\Lambda$ be a lattice such that $K+\Lambda$ is a minimal covering of $\mathbb{R}^n$. Suppose that $K$ contains a translate of a full-dimensional lattice polytope $Q$, so that

$$K-\mathbf{x}\supset Q$$

for some $\mathbf{x}\in\mathbb{R}^n$.

Boundary-vertex conjecture. The vertices of $Q$ lie on the boundary of $K-\mathbf{x}$.

The conjecture proposes a stronger boundary constraint for lattice polytopes contained in minimal covering bodies, following the paper's proof that minimal covering bodies are polytopes.