Amending the Lonely Runner Spectrum Conjecture

Let $\mathrm{\mid }\mathrm{\mid }x\mathrm{\mid }\mathrm{\mid }$ be the absolute distance from $x$ to the nearest integer. For a set of distinct positive integral speeds $v_1,\dots ,v_n$, we define its maximum loneliness, also known as the gap $\delta$, to be $$ML(v_1,\dots ,v_n)=\underset{t\in \mathbb{R}}{\max }\underset{1\le i\le n}{\min }\mathrm{\mid }\mathrm{\mid }t{v}_i\mathrm{\mid }\mathrm{\mid }\mathrm{.}$$ The Loneliness Spectrum Conjecture, recently proposed by Kravitz (2021), asserts that $$\mathrm{\exists }s\in \mathbb{N},\text{ML}(v_1,\dots ,v_n)=\frac{s}{sn+1}\text{ or ML}(v_1,\dots ,v_n)\ge \frac{1}{n}\mathrm{.}$$ We disprove the Loneliness Spectrum Conjecture for $n=4$ with an infinite family of counterexamples and propose an alternative conjecture. We confirm the amended conjecture for $n=4$ whenever there exists a pair of speeds with a common factor of at least $3$ and also prove some related results.