Intermediate-step conjectures for the logarithmic Kazhdan–Lusztig correspondence

Let $\mathcal{W}^{\log}$ be the proposed logarithmic conformal field theory obtained from the screening-kernel construction, and let lattice-VOA modules be restricted to it. Intermediate correspondence conjectures. The irreducible $\mathcal{W}^{\log}$-modules should arise as unique irreducible quotients of such restricted lattice-VOA modules; Weyl-group or Weyl-groupoid combinations of short screenings should act on lattice-VOA representations and intertwine the relevant Virasoro actions; these screening maps should determine the decomposition of restricted modules; the resulting irreducibles should have nonzero $\operatorname{Ext}$ groups precisely along Weyl-group orbits; and one should be able to construct a bimodule from the reflected lattice VOAs that is a projective cover of the vacuum representation and carries a full $u_q(\mathfrak{g})$-action. These are proposed steps toward the correspondence, and the source gives no resolution.