The additive complexity conjecture for automatic sequences

Let $k\geq 2$ be an integer, and let $\mathbf{w}$ be a $k$-automatic sequence over a finite set of non-negative integers. Its additive complexity is the function assigning to each positive integer $n$ the number of distinct sums of length-$n$ factors of $\mathbf{w}$.

Additive complexity conjecture. The additive complexity of a $k$-automatic sequence is a $k$-regular sequence.

This conjecture extends the analogous conjecture for abelian complexity. The paper proves a particular case, while the general statement remains open.