Finite generation conjecture for invariant jet polynomials under reparametrization

Let $G_k$ be the group of $k$-jet reparametrizations acting on the space of $k$-jets $J_k\mathbb{C}^n$, with coordinates represented by $f'(0),f”(0),\ldots,f^{(k)}(0)$. The induced action of $G_k$ on the polynomial ring of these jet variables defines the invariant ring

$$\mathcal{O}(J_k \mathbb{C}^n)^{G_k}=\mathbb{C}[f'(0),f”(0),\ldots,f^{(k)}(0)]^{G_k}.$$

Finite generation conjecture. The ring of invariant polynomials

$$\mathcal{O}(J_k \mathbb{C}^n)^{G_k}=\mathbb{C}[f'(0),f”(0),\ldots,f^{(k)}(0)]^{G_k}$$

is finitely generated.

Determining finite generation for these complex graded invariant algebras has long been an open problem in the area, and the conjecture proposes finite generation for the invariant polynomials of the non-reductive reparametrization group $G_k$.