Free-field realization conjecture for W-algebras

Let $\Lambda^{long}$ be the non-fractional lattice, and let $zem_{\alpha_i^{short}}$ and $zem_{\alpha_i^{long}}$ be the short and long screening operators. Define

$$\mathrm{W}:=\mathcal{V}_{\Lambda^{long}}\;\cap\; \bigcap_i\ker(\zemlja_{\alpha_i^{short}})\;\cap\;\bigcap_i\ker(\zemlja_{\alpha_i^{long}}).$$

Free-field realization conjecture. In the Lie-algebra case, $\mathrm{W}$ should be isomorphic to the corresponding W-algebra, namely the Hamiltonian or quantum Drinfeld–Sokolov reduction of the affine Lie algebra $\widehat{\mathfrak{g}}$ at level $ell$. This realization is stated to hold for generic $q$ and to be widely believed otherwise, but the non-generic case is described as a difficult open conjecture.