Conjecture on finite-multiplicity representations of minimal W-algebras

Let $\mathfrak g$ be the Lie superalgebra under consideration, let $k$ lie in its unitarity range, and set

$$\mathcal E_k^{\mathrm{fin}}\subset \mathcal E(W_k^{\min}(\mathfrak g))$$

to be the subcategory of weight $W_k^{\min}(\mathfrak g)$-modules with finite multiplicities. Here a weight module is a module that is a weight module for $\widehat{\mathfrak g}^{\natural}$. Finite-multiplicity conjecture. The irreducible highest weight $W_k^{\min}(\mathfrak g)$-modules are precisely all the irreducible representations in the category $\mathcal E_k^{\mathrm{fin}}$. The conjecture asks whether the finite-multiplicity weight-module category contains no irreducible modules beyond the irreducible highest weight ones; the source presents it as an open question.