Conjecture on the string attractor size of simple-Parry automatic sequences

Let $c$ be a finite word satisfying condition (WH), and assume that one of the equivalent assertions in Proposition 4 holds. Let $\mathbf{u}$ be the associated simple-Parry automatic sequence, let $k$ be the length parameter from that proposition, and let $U_i$ denote the sequence of lengths associated with $c$. The function $s_{\mathbf{u}}(m)$ is the minimum size of a string attractor of the length-$m$ prefix of $\mathbf{u}$. String-attractor-size conjecture.

$$s_{\mathbf{u}}(m)=\begin{cases} i+1, & \text{if }m\in[U_i,U_{i+1})\text{ with }0\leq i\leq k-2,\\ k, & \text{if }m\geq U_{k-1}. \end{cases}$$

This conjecture predicts the exact minimal string-attractor size for the prefixes of the sequences under consideration, extending the known behavior of the period-doubling word and the bounds established earlier in the paper. Its status is not resolved in the supplied source.