Automaticity of number walls of automatic sequences

Establish that for every automatic sequence S over a finite field F_q, the number wall W_q(S) (the two-dimensional array of Toeplitz determinants associated to S over F_q) is itself a two-dimensional automatic sequence.

Background

Several classical automatic sequences (Thue–Morse, Paperfolding, Cantor) have number walls whose profiles are known to be two-dimensional automatic sequences. This empirical evidence led to a general conjecture posed by Garrett and the second named author that automaticity of a one-dimensional sequence should be inherited by its number wall.

The present paper proves new cases (notably for the p-Cantor sequence), giving further evidence toward the conjecture, but the general statement remains open and is stated explicitly as a conjecture.