Garrett–Rogers conjecture on automaticity of number walls

Let $q$ be a positive power of a prime, and let $\textbf{S}$ be an automatic sequence over $\mathbb{F}_q$. Let $W_q(\textbf{S})$ denote the number wall associated with $\textbf{S}$. Garrett–Rogers conjecture. The number wall $W_q(\textbf{S})$ is itself a two-dimensional automatic sequence. Number walls of several familiar automatic sequences are known to be two-dimensional automatic, motivating this general statement; the source provides no resolution status for the conjecture.