19.4 Separating words problem

Given two words \(u, v\) of length \(n\) over \(\{a, b\}\). Does there always exist an automaton of size \(O(\log n)\) which distinguishes \(u\) and \(v\). Or bigger, like \(O(n^{1/3})\)? Best known bound is about \(O(n^{2/5} \log(n))\).