The fact is that any crosswind component means you fly a longer distance, unless there is a tailwind component to it which is sufficient to compensate. So even a perfect 90 degree crosswind means you have to fly further than the GC route.
It thus follows that if you get a perfect 90 deg c/w from the right and then get a perfect 90 deg c/w from the left, you end up flying a longer distance to get there.
I don't know the answer, and I am too lazy to try some trig, but I think just following the magenta line (and thus crabbing the wind) will be most efficient because the aircraft is flying the shortest route and thus the wind has the least opportunity to "work" on it.
WN's suggestion of a constant zero-wind heading looks tempting and could well be right, but I feel that the aircraft will be subjected to the effect of the wind for longer.