Is it true that if $A \subseteq \mathbb{N}\setminus\{1\}$ is a finite set with $\sum_{n \in A} \frac{1}{n} < 2$ then there is a partition $A=A_1 \sqcup A_2$ such that $\sum_{n \in A_i} \frac{1}{n} < 1$ for $i=1,2$?

This is not true in general, as shown by Sándor [Sa97], who observed that the proper divisors of $120$ form a counterexample. More generally, Sándor shows that for any $n\geq 2$ there exists a finite set $A\subseteq \mathbb{N}\backslash\{1\}$ with $\sum_{k\in A}\frac{1}{k} < n$ and no partition into $n$ parts each of which has $\sum_{k\in A_i}\frac{1}{k}<1$.