The restricted amended Loneliness Spectrum Conjecture for four speeds

Let $v_1,v_2,v_3,v_4$ be positive integers, and let

$$\operatorname{ML}(v_1,v_2,v_3,v_4)=\max_{t\in\mathbb{R}}\min_{1\leq i\leq 4}\lVert tv_i\rVert,$$

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

$$\exists s,k\in\mathbb{N},\quad k\in\{1,2\},\quad \operatorname{ML}(v_1,v_2,v_3,v_4)=\frac{s}{4s+k},$$

or

$$\operatorname{ML}(v_1,v_2,v_3,v_4)=\frac{1}{5}.$$

The claim is motivated by computational evidence for four speeds: values with $k\notin\{1,2\}$ were not found below the stated bound. The source presents it as a further conjecture, without a general proof.