General Noetherianity conjecture for pointwise finitely generated $VIC(\mathbb{Z})$-modules

A $VIC(\mathbb{Z})$-module is a functor from the category $VIC(\mathbb{Z})$; it is finitely generated if it is generated by finitely many elements under the morphisms of this category, and pointwise finitely generated over a Noetherian ring $R$ if each value is a finitely generated $R$-module. General Noetherianity conjecture. If $V$ is a finitely generated $VIC(\mathbb{Z})$-module which is pointwise finitely generated over a Noetherian ring $R$, then every $VIC(\mathbb{Z})$-submodule of $V$ is also finitely generated. This would extend the known Noetherianity results over $\mathbb{C}$ and over finite rings and fields of positive characteristic; the conjecture is stated as a broader expected generalization.