The Loneliness Spectrum Conjecture

Let $v_1,\ldots,v_n$ be positive integers, and let

$$\operatorname{ML}(v_1,\ldots,v_n)=\max_{t\in\mathbb{R}}\min_{1\leq i\leq n}\lVert tv_i\rVert,$$

where $\lVert x\rVert$ denotes the absolute distance from $x$ to the nearest integer. The Loneliness Spectrum Conjecture. One has either

$$\exists s\in\mathbb{N},\quad \operatorname{ML}(v_1,\ldots,v_n)=\frac{s}{ns+1},$$

or

$$\operatorname{ML}(v_1,\ldots,v_n)\geq\frac{1}{n}.$$

This conjecture refines the Lonely Runner Conjecture by describing the possible values below $1/n$. Kravitz proved it for $n=3$, but the paper gives counterexamples for $n=4$ and $n=6$, so the conjecture is refuted.