Characterize 3-manifolds admitting stably exotic fillings

Characterize which closed, smooth 3-manifolds admit stably exotic fillings, meaning pairs of fillings that are homeomorphic relative to the boundary but not stably diffeomorphic relative to the boundary.

Background

The paper establishes sufficient conditions for a closed, smooth 3-manifold to admit stably exotic fillings, including orientability, being a circle bundle over a closed surface other than $RP^2$, and admitting a null-bordant tangential Pin$^+$ structure. It also proves that closed 3-manifolds containing a two-sided $RP^2$ do not admit such fillings. These results provide only a partial characterization, leaving the general classification problem unresolved.