A supporter of the globe, and (a supporter) of the flat earth theory, go on a walk together...

….around the globe/across the plane.
They go West, from a certain spot in Europe, and they’ll remain moving that direction until they come back to that exact point (whatever means of transportation they need to use when they can’t walk will also keep going West).

Who is going to be red-faced when they’ve arrive back at the start/finish line?

(Red-faced, not because of exhaustion, but of embarrassment).

Or, “Will this walk be a sufficient way to convince the other party of their being wrong”?

Will the ‘loser’ (whoever that may be) be now convinced that they’ve been wrong?