The flatness–word-problem characterization conjecture

Let $G$ be a finitely generated group, and let $\mathbf{C}$ be a class of languages closed under inverse homomorphism.

Flatness–word-problem conjecture. $G$ is $\mathbf{C}^\forall$-flat if and only if the word problem of $G$ lies in $\mathbf{C}$.

This conjecture would connect the notion of $\mathbf{C}^\forall$-flatness with the word problem. It is refuted: the classes of poly-context-free, context-sensitive, and recursive languages provide counterexamples.