Finite-generation conjecture for modules over the completed algebra of vector fields

Let $X$ be a quasi-projective variety, let $U$ be an affine open subset, and write $A={\mathcal A}(U)$ and $AV={\mathcal {AV}}(U)$. Let $M$ be an $AV$-module that is finitely generated as an $A$-module, and let $J_m$ denote the corresponding level-$m$ ideal in $AV$. Finite-generation conjecture. There exists $m\in\mathbb N$, depending on the rank of $M$, such that $J_m$ acts trivially on $M$. This would make the completion redundant for finite-rank modules, allowing one to work with a quotient algebra; the source further suggests consequences for local descriptions on étale charts and for sheafifying modules over smooth affine varieties, but does not establish the conjecture.