Orientability of components of the Fisk complex

Determine whether, for a 4-manifold and a triangle set T whose boundary edges each belong to at least two triangles, some 2-manifold components generated by T can be orientable while other components are nonorientable.

Background

For a 4-manifold, the paper considers collections of triangles with the property that every boundary edge has at least two attached triangles. Such a collection generates a simplicial complex decomposable into finitely many 2-manifolds whose pairwise intersections are empty or curves. The paper observes that the Euler characteristic of the collection can be related to the Euler characteristics of these components, but leaves unresolved whether different components in the same decomposition can have different orientability types.