Projectivity and suitability of O_{c_{p,q}}^0 for logarithmic minimal models

Show that the braided tensor subcategory O_{c_{p,q}}^0 of Virasoro modules at central charge c_{p,q} (consisting of modules that induce to untwisted W_{p,q}-modules) has enough projective objects and determine that O_{c_{p,q}}^0 is the correct Virasoro module category for constructing W-extended logarithmic minimal model conformal field theories.

Background

The authors construct and paper the subcategory O_{c_{p,q}}^0 of Virasoro modules at central charge c_{p,q}, defined by induction to ordinary (untwisted) W_{p,q}-modules. They prove several structural properties (tensor embedding, closure under contragredients) and position O_{c_{p,q}}^0 as a candidate framework for logarithmic minimal models.

They explicitly conjecture that O_{c_{p,q}}^0 has enough projective objects and serves as the appropriate Virasoro module category for building full logarithmic minimal model CFTs, emphasizing that these points remain unresolved and central for future development of the theory.