Let $U=\{\frac{1}{n}: n\in\mathbb{N}\} \cup \{-\frac{1}{n}: n\in\mathbb{N}\}$ be the set of positive or negative unit fractions.

Are there positive integers $m<n \in \mathbb{N}$ such that for every $u,v\in U$ there are $u',v'\in U$ such that $\frac{m}{n} - |u'+v'| < \frac{m}{n} - |u+v|$?