Let $\mathbb{N}$ denote the set of positive integers. For every integer $k\in\mathbb{N}$ let $m(k)$ denote the minimal size of a finite set $S\subseteq \mathbb{N}$ such that $\sum_{j\in S}j^{-1}=k$.

Is there $z\in \mathbb{N}$ such that $m(k) = {\cal O}(k^z)$?