Zonotope reformulation of the Lonely Runner Conjecture for $T_{\mathbf n}$

Let $\mathbf{n}\in\mathbb{Z}^d_{>0}$ be a velocity vector, and let

$$T_{\mathbf n}=\operatorname{\mathcal{Z}}(\mathbf e_1,\ldots,\mathbf e_d,3\mathbf n).$$

Here an interior lattice point means a point $\mathbf w\in\operatorname{int}(T_{\mathbf n})\cap\mathbb{Z}^d$, and $\operatorname{ca}(T_{\mathbf n},\mathbf w)$ is its coefficient of asymmetry. Zonotope reformulation. For every velocity vector $\mathbf n\in\mathbb{Z}^d_{>0}$, there exists an interior lattice point $\mathbf w\in\operatorname{int}(T_{\mathbf n})\cap\mathbb{Z}^d$ such that

$$\operatorname{ca}(T_{\mathbf n},\mathbf w)\leq d.$$

The paper states that this is equivalent to the Lonely Runner Conjecture after replacing the unbounded zonohedron by the suitable zonotope $T_{\mathbf n}$. It is consequently open in general.