Coefficient-of-asymmetry reformulation of the Lonely Runner Conjecture

Let $\mathbf{n}\in\mathbb{Z}^d_{>0}$ be a velocity vector, meaning that its entries are distinct and relatively prime, and let

$$Z(\mathbf{n})=\mathbb{R}\mathbf{n}+[0,1]^d.$$

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

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

This is presented as an equivalent geometric formulation of the Lonely Runner Conjecture, via the identity between the minimum coefficient of asymmetry and the gap of loneliness. Its status is therefore the same as that conjecture and remains open in general.