Wills's version of the Lonely Runner Conjecture

Let $k\ge 2$ be an integer, and let $m_1,\ldots,m_k$ be distinct positive integers. For $x\in\mathbb{R}$, let $\|x\|$ denote the distance from $x$ to the nearest integer:

$$\|x\|=\min\{|x-n|:n\in\mathbb{Z}\}.$$

Wills's conjecture. If $1\le m_1<m_2<\ldots<m_k$, then

$$\max_{\alpha\in\mathbb{R}}\min_{1\le j\le k}\|m_j\alpha\|\ge\frac{1}{k+1}.$$

This is the Diophantine-approximation formulation of the Lonely Runner Conjecture, originating independently with Wills and Cusick. The supplied source does not state whether the conjecture is resolved.