Let $C>0$. Does there exist a $c>0$ (depending on $C$) such that, for all sufficiently large $x$, if $A\subseteq [1,x]$ has ${\sum }_{n\in A}\frac{1}{n}\le C$ then$$\#\{m\le x:a\nmid m\text{ for all }a\in A\}\gg \frac{x}{(\log x{)}^c}?$$