The algebraicity criterion for generalized double Laurent series

Let $p$ be prime, let $q$ be a power of $p$, and let $S_p$ denote the nonnegative integers whose base-$p$ expansions are used in the definition of $p$-automatic functions. Let $f:\mathbb{Q}\times\mathbb{Q}\to\mathbb{F}_q$ have well-quasi-ordered support $S$, and write

$$\sum_{i,j}f(i,j)t^iu^j\in\mathbb{F}_q((t^{\mathbb{Q}},u^{\mathbb{Q}})).$$

For positive integers $a,b$, set

$$aS+b=\{(ai+b,aj+b):(i,j)\in S\},$$

and, when $aS+b\subseteq S_p\times S_p$, define

$$f_{a,b}(x,y)=f\left(\frac{x-b}{a},\frac{y-b}{a}\right).$$

Algebraicity criterion. The generalized double Laurent series is algebraic over $\mathbb{F}_q(t,u)$ if and only if: (a) for some positive integers $a$ and $b$, $aS+b\subseteq S_p\times S_p$; and (b) for some $a,b$ satisfying (a), $f_{a,b}:S_p\times S_p\to\mathbb{F}_q$ is $p$-automatic. Moreover, when these conditions hold, $f_{a,b}$ is $p$-automatic for every pair of nonnegative integers $a,b$ for which (a) holds.

This conjecture proposes a two-variable analogue of the algebraicity criterion for generalized power series over finite fields, extending the relationship between algebraicity and automaticity to well-quasi-ordered supports in $\mathbb{Q}\times\mathbb{Q}$. Its status is not established by the supplied source context.