The nonchalant-word infinitude conjecture

Fix an ordered alphabet $\mathbb A_n=\{\mathtt{1,2,\dots,n}\}$. Define nonchalant words recursively by $N_1=\mathtt 1$ and $N_{i+1}=N_i'xN_i”$, where this is a square-free extension of $N_i$, $N_i”$ is the shortest possible suffix of $N_i$, and $x\in\mathbb A_n$ is the earliest possible letter.

Nonchalant-word infinitude conjecture. The sequence of nonchalant words over $\mathbb A_n$ is infinite for every $n\geqslant 3$.

The conjecture asserts that the greedy construction never stops. Computer experiments support it, including a ternary nonchalant word of length $10000$, but the general claim remains open.