Better-quasi-order conjecture for continuous rational-valued functions

Let $[\omega]^2_+$ denote the relevant domain of ordered pairs of natural numbers, let $C([\omega]^2_+,\mathbb{Q})$ be the space of continuous functions from this domain to $\mathbb{Q}$, and let $\sqsubseteq$ denote embeddability. Better-quasi-order conjecture. The quasi-order

$$(C([\omega]^2_+,\mathbb{Q}),\sqsubseteq)$$

is better-quasi-ordered. This conjecture is proposed as an essential step toward removing the local compactness assumption on the domain in the paper's dichotomy; its resolution is not given here.