Two different surfaces, a flat disk and a tall dome, share the same boundary circle $C$. How do the fluxes of $\nabla\times\vec{F}$ through them compare?

B. They are equal: both tie to the same line integral

Curl flux is determined entirely by the rim: any surface spanning $C$ — disk, dome, or crumpled sheet — yields the identical value. A soap film can bulge freely on a wire loop without changing the circulation around the wire.