The weak Loneliness Spectrum Conjecture for one very fast speed

Let $n\geq4$ be an integer. For positive integers $v_1<\cdots<v_n$, define

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

Weak Loneliness Spectrum Conjecture. For every integer $n\geq4$, there exists a function $f_n:\mathbb{N}\to\mathbb{N}$ such that, whenever $v_n>f_n(v_{n-1})$, either

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

or

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

This is a regime-specific refinement of the Loneliness Spectrum Conjecture, addressing sets of speeds with one speed sufficiently larger than the preceding one. The paper proves a stronger explicit bound for $n=4$ and $n=6$, but the assertion for every $n\geq4$ remains open.