Lower-boundedness of irreducible U^pr-modules

Show that every irreducible module over the principal W-algebra U^pr of V_k(psl(2|2)) is lower bounded in conformal weight, thereby confirming that all irreducible U^pr-modules are ordinary.

Background

The authors compute the Zhu algebra of U^pr and prove that all irreducible lower-bounded U^pr-modules are highest weight with finite-dimensional top spaces. They expect all irreducible modules to be ordinary (lower bounded), but only prove results under a lower-boundedness hypothesis.

Establishing lower-boundedness in full generality would complete the picture of ordinary representation theory for U^pr and align it with principal W-algebras associated to Lie algebras.