Erdős problem 206

Let $x>0$ be a real number. For any $n\geq 1$ 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?)

Kovač [Ko24b] has proved that this is false - in fact as false as possible: the set of $x\in (0,\infty)$ for which the best underapproximations are eventually 'greedy' has Lebesgue measure zero.