Finite-generation conjecture for jet-differential invariant fibres

Let $J_{k,p}$ denote the bundle whose fibre at $x$ is $(J_{k,p})_x$, and let $E_{k,p}^n$ be the associated bundle. For $p\geq 1$ and $k$ sufficiently large depending on $p$, consider the invariant fibre algebras

$$\mathcal{O}((J_{k,p})_x)^{\mathbb{G}_{k,p}'}.$$

Finite-generation conjecture. Every fibre $\mathcal{O}((J_{k,p})_x)^{\mathbb{G}_{k,p}'}$ of $E_{k,p}^n$ is a finitely generated graded complex algebra.

This is stated as a consequence of the preceding finite-generation conjecture and would provide finite generation of the graded algebras of invariant jet differentials.