Error in Robson's proof about separating strings?

Question

One of my students discovered a possible mistake in Robson's classic paper Separating strings with small automata.
The issue is in the proof of Theorem 1, giving the simpler bound $O(\sqrt{n\log n})$.
The proof goes by giving an automaton that finds a substring of length $O(\sqrt{n\log n})$ in $u$ that ends in $i$ such that $u_i\ne v_i$ is the first place where the words $u$ and $v$ differ.
It is correctly proved that the given substring cannot occur in $v$ up to $i$, starting with some required modulo, but the issue is that the substring can arise after $i$.
I think that he is right, and see no way of saving the proof.

ps. Note that the best claimed bound in Robson is $\tilde O(n^{2/5})$ recently a better bound, $\tilde O(n^{1/3})$ was proved by Chase.

Answer 1

Converting my comment to an answer:

This is taken care of in the paragraph just before Theorem 1. If the accepting state is reached at some $v_j$, then we necessarily have $j>i$, and we can compose the automaton with one that separates $v_{j+1}\ldots v_n$ and $u_{i+1}\ldots u_n$, which have different lengths so are easy to separate.