The lonely runner conjecture for integer speed sets

Let $A\subset\mathbb{Z}_{>0}$ be a set of size $n$, and let $\mathbb{T}=\mathbb{R}/\mathbb{Z}$. For $t\in\mathbb{R}$, write $\lVert t\rVert$ for the distance from $t$ to the nearest integer.

Lonely runner conjecture. There exists $x\in\mathbb{T}$ such that

$$\frac{1}{n+1}\leq\lVert ax\rVert\leq1-\frac{1}{n+1}$$

for every $a\in A$.

This is the formulation of the lonely runner conjecture arising from the paper's study of sets of the form $\{u,2u,v,2v\}$. Its general status is open.