Non-finite generation of semirings of p-cardinalities

Let $\mathcal{C} \supseteq \mathsf{P}$ be a standard complexity class, such as $\mathsf{P}$, $\mathsf{NP}$, $\mathsf{PSPACE}$, or $\mathsf{EXP}$. Consider the semiring formed by the p-cardinalities of languages in $\mathcal{C}$, together with the corresponding semiring of equivalence classes of p-cardinalities.

Non-finite-generation conjecture. For every such complexity class $\mathcal{C}$, the semiring of p-cardinalities of languages in $\mathcal{C}$ is not finitely generated. The same holds for the semiring of equivalence classes of p-cardinalities.

This predicts a strong algebraic complexity phenomenon even within standard complexity classes, including $\mathsf{P}$, $\mathsf{NP}$, $\mathsf{PSPACE}$, and $\mathsf{EXP}$. The supplied text gives no resolution, so both non-finite-generation assertions remain open.