The nonchalant-word infinitude conjecture for three-letter alphabets

Let $\mathbb A$ be a fixed ordered alphabet, and recursively construct the sequence of nonchalant words by starting with $G_1=\mathtt a$ and, at each step, inserting the earliest possible letter in the rightmost possible position so that the resulting word remains square-free. Nonchalant-word infinitude conjecture. The sequence of nonchalant words over a $3$-letter alphabet is infinite. This conjecture concerns whether the greedy square-free extension procedure ever stops; the paper gives initial examples and motivates the question through a recursive construction based on computer verification.