Crosscap genus in lens spaces

Determine whether the crosscap genus of the projection of the great circle C to the lens space L(p,q) is realized by the particular surfaces \tilde{\xi}_{k,m} enumerated in Proposition \ref{crosscap}.

Background

For odd p, the paper constructs non-orientable minimal fillings in lens spaces and identifies explicit surfaces descending from the \tilde{\xi}_{k,m} family. The crosscap genus is the least genus of an embedded non-orientable surface bounding the projected curve, and it is unresolved whether the explicitly constructed candidates attain this minimum.