The finite-generation conjecture for gauge modules

Let $X$ be an affine variety with coordinate algebra $A_X$, let $A\mathcal{V}_X$ denote the corresponding algebra of functions and vector fields, and let a gauge module mean an $A\mathcal{V}_X$-module locally realized as an $A$-submodule of $A_{(h)}\otimes U$ for a finite-dimensional module $U$ of the Lie algebra of vector fields of non-negative degree, with the action involving gauge fields. Gauge-module conjecture. Every $A\mathcal{V}_X$-module that is finitely generated over $A_X$ is a gauge module. This would indicate that gauge modules form a broad class encompassing all finitely generated modules over the algebra of functions and vector fields; the paper provides the conjecture but no resolution is stated in the supplied text.