On eventually greedy best underapproximations by Egyptian fractions

Erd{o}s and Graham found it conceivable that the best n-term Egyptian underapproximation of almost every positive number for sufficiently large n gets constructed in a greedy manner, i.e., from the best $(n-1)$-term Egyptian underapproximation. We show that the opposite is true: the set of real numbers with this property has Lebesgue measure zero. [This note solves Problem 206 on Bloom's website "Erd{o}s problems".]