Open. The site labels the problem OPEN. Two results are in hand: Erdős's lower bound $\epsilon_n\gg\log\log\log n/\log\log n$, asserted in [Er79d] p. 79 ("I can prove") and [Er80] p. 106 ("It is not difficult to prove") without a proof in either source, which the formal-conjectures collection marks research solved as a variant, and the elementary counting bound $\limsup\epsilon_n\le1/e$ proved under What is known. No upper bound tending to $0$, and nothing toward $\epsilon_n=o(1)$, was found in the search whose scope the Current assessment records. Erdős's own remark in [Er80] ties the question to the Erdős--Ruzsa covering conjecture of Problem 1200: if that conjecture holds "then very likely $\epsilon_x>c$ for some absolute constant $c$", that is, the answer to the displayed question would be no. This is a bounded negative finding, not a certificate of openness.