OPEN

What is the size of the largest $A\subseteq \{1,\ldots,N\}$ such that for any $a,b_1,\ldots,b_k\in A$ with $k\geq 2$ we have
\[\frac{1}{a}\neq \frac{1}{b_1}+\cdots+\frac{1}{b_k}?\]
Is there a constant $c>1/2$ such that $\lvert A\rvert >cN$ is possible for all large $N$?

The example $A=(N/2,N]\cap \mathbb{N}$ shows that $\lvert A\rvert\geq N/2$.