The nonchalant-word convergence conjecture

Fix an ordered alphabet $\mathbb A_n=\{\mathtt{1,2,\dots,n}\}$ and let $(N_i)$ be the sequence of nonchalant words produced by the recursive square-free greedy procedure.

Nonchalant-word convergence conjecture. The sequence of nonchalant words over $\mathbb A_n$ converges to an infinite word $\mathcal{N}_n$ for every $n\geqslant 3$.

The paper presents this as a plausible strengthening suggested by experiments; no proof or resolution is supplied.