Fan–Sun amended Loneliness Spectrum Conjecture

For a positive-integer speed set $V$, let $\kappa(V)$ denote its gap of loneliness, and let $n=|V|$. Amended Loneliness Spectrum Conjecture. For every $n\in\mathbb{N}$ and every $n$-set $V$ of positive integers, either

$$\kappa(V)=\frac{s}{sn+k}\quad\text{for some }s,k\in\mathbb{N},$$

or

$$\kappa(V)\geq\frac{1}{n}.$$

This is a weakening proposed by Fan and Sun after Kravitz's original spectrum conjecture was disproved for $n=4$ (and similarly for $n=6$). The source presents it as an open conjecture.