The cut-support conjecture for mixed Witt algebra modules

Let $\mathfrak{g}$ be the Lie algebra under consideration, graded by $\mathbb{Z}^n$, and let $V$ be a mixed $\mathfrak{g}$-module: there are weights $\lambda$ and $\lambda+\alpha$ such that one corresponding weight space is infinite-dimensional and the other finite-dimensional. In the terminology of the source, a module is cut when its support has the associated cut-type form. Cut-support conjecture. Any mixed $\mathfrak{g}$-module is cut. This proposed classification of mixed-module supports follows the discussion of mixed dense and mixed punctured modules and is intended to describe the support behavior beyond the known Virasoro case.