The nonexistence conjecture for extremal square-free words over four-letter alphabets

A square-free word is a word containing no factor of the form $uu$ with $u$ nonempty, and an extremal square-free word is a square-free word that cannot be extended by inserting a single letter in any position while remaining square-free. Extremal-word nonexistence conjecture. There are no extremal square-free words over a $4$-letter alphabet. The conjecture is motivated by a computer search that found no extremal words over four letters of length up to $1000$. If true, every square-free word over four letters could be extended by inserting a letter, and this would imply the corresponding infinitude conjecture for nonchalant words over four-letter alphabets.