SOLVED

Let $\alpha >0$ and $N\geq 1$. Is it true that for any $A\subseteq \{1,\ldots,N\}$ with $\lvert A\rvert \geq \alpha N$ there exists some $S\subseteq A$ such that
\[\frac{a}{b}=\sum_{n\in S}\frac{1}{n}\]
with $a\leq b =O_\alpha(1)$?

Liu and Sawhney [LiSa24] observed that the main result of Bloom [Bl21] implies a positive solution to this conjecture. They prove a more precise version, that if $(\log N)^{-1/7+o(1)}\leq \alpha \leq 1/2$ then there is some $S\subseteq A$ such that
\[\frac{a}{b}=\sum_{n\in S}\frac{1}{n}\]
with $a\leq b \leq \exp(O(1/\alpha))$. They also observe that the dependence $b\leq \exp(O(1/\alpha))$ is sharp.