Stratification of the category of graded current-algebra modules

Let $\mathbf{g}$ be a finite-dimensional Lie algebra, and let $\mathcal{I}(\mathbf{g}\otimes\mathbb{C}[x])$ denote the category of graded finitely generated $\mathbf{g}$-locally finite modules over the current Lie algebra $\mathbf{g}\otimes\mathbb{C}[x]$. Current-algebra stratification conjecture. The category $\mathcal{I}(\mathbf{g}\otimes\mathbb{C}[x])$ is stratified. This is presented as a principal application of the general theory; the supplied text describes known BGG reciprocity results in special cases but gives no resolution of the stated general claim.