Lengths of extremal square-free ternary words

A square-free word w over a fixed alphabet Sigma is extremal if every word obtained from w by inserting a single letter from Sigma (at any position) contains a square. Grytczuk et al. recently introduced the concept of extremal square-free word and demonstrated that there are arbitrarily long extremal square-free ternary words. We find all lengths which admit an extremal square-free ternary word. In particular, we show that there is an extremal square-free ternary word of every sufficiently large length. We also solve the analogous problem for circular words.