We say that $X$ is a linearly ordered topological space (LOTS) if $X$ has the order topology induced by a linear order on $X$.
On the other hand, we say that $X$ is a shrinking space if every open cover has a shrinking, i.e., if for every open cover $\mathcal{U}$ of $X$ there exists an open refinement $\mathcal{V}=\{V_U:U\in\mathcal{U}\}$ such that $\overline{V_U}\subset U$ for all $U\in\mathcal{U}$.
While searching about shrinking spaces on the web, I came across this note, where the author claims that every LOTS is a shrinking space.
Since I was not able to prove this fact, I did some digging. However, the only reference I found is a paper from Fleischman (On coverings of linearly ordered spaces, 1970), which I am not able to get.
So I would like to ask for hints on how to prove that every LOTS is a shrinking space, or at least another reference where the proof can be found.
Every LOTS is monotonically normal, and as such normal and countably paracompact. The note you quoted proves that this implies that a LOTS is shrinking (we only need normal and countably metacompact).