The regularity conjecture for the $k$-abelian complexity of automatic sequences

Let $k\geq 1$ and let $\omega$ be an $\ell$-automatic sequence, meaning that its $\ell$-kernel is finite. The $k$-abelian complexity of $\omega$ is the sequence counting the $k$-abelian equivalence classes of factors of each length.

Automatic-sequence regularity conjecture. The $k$-abelian complexity of any $\ell$-automatic sequence is an $\ell$-regular sequence.

The conjecture generalizes the previously conjectured—and independently proved—case of the $2$-abelian complexity of the Thue–Morse sequence. The paper presents the $3$-regularity of the $k$-abelian complexity of the Cantor sequence as supporting evidence; the general assertion remains open in the supplied source.