Conjecture that long screenings represent the corresponding Lie algebra

Let $zem_{\alpha_i^{long}}$ be the long screening operators associated with the long basis of the integral lattice, and let $\mathfrak{g}$ denote the corresponding Lie algebra (or the dual root system or Cartan-type Lie algebra in the stated generalizations). Long-screening representation conjecture. The long screenings should constitute a representation of the corresponding Lie algebra, commuting with the Virasoro action. In the deformed lattice VOA $\mathcal{V}_\Lambda^{(\Omega,1)}$, the resulting Lie-algebra structure may involve commutators and anticommutators. The simply-laced case is stated to be proved; the general case remains conjectural.