The amended 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 amended Loneliness Spectrum Conjecture. One has either

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

or

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

This modification is proposed after the original spectrum conjecture is disproved by examples such as $\operatorname{ML}(8,3,11,19)=7/30$ and $\operatorname{ML}(5,6,11,17,23,28)=8/51$. The paper reports experimental support and proves special cases, but does not establish the amended conjecture in general.