Let $x>0$ be a real number. For any $n\ge 1$ let$$R_n(x)=\sum _{i=1}^n\frac{1}{m_i}<x$$be the maximal sum of $n$ distinct unit fractions which is $<x$.Is it true that, for almost all $x$, for sufficiently large $n$, we have$$R_{n+1}(x)=R_n(x)+\frac{1}{m},$$where $m$ is minimal such that $m$ does not appear in $R_n(x)$ and the right-hand side is $<x$? (That is, are the best underapproximations eventually always constructed in a 'greedy' fashion?)