On the intervals for the non-existence of covering systems with distinct moduli

In a research seminar in $2006$, M. Filaseta, O. Trifonov, and G. Yu showed for each integer $n\ge 3$ there is no distinct covering with all moduli in the interval $[n,6n]$. In $2022$, this interval was subsequently improved to $[n,8n]$ by the first author and O. Trifonov. The first author then improved this bound to $[n,9n]$ in $2023$. Building off their method, we show that for each integer $n\ge 3$, there does not exist a distinct covering system with all moduli in the interval $[n,10n]$.