The generalized nonchalant-word infinitude conjecture

Let $\ae$ denote the function used in the paper to measure the relevant extension parameter for words, and let $N_1$ be the starting word of the nonchalant algorithm.

Generalized nonchalant-word infinitude conjecture. If $\ae(N_1)>2$ for the starting word $N_1$ of the nonchalant algorithm, then the respective sequence of nonchalant words is infinite.

This is presented as a version of the nonchalant-word conjecture motivated by numerical experiments. Because the supplied context does not define $\ae$, the precise scope of the hypothesis should be checked against the paper.