I just had to try it for myself.....
I set things up, pretty much as PropStrike discribed in his flight, except, I used my C-5 Galaxy. I've done some tweaking to her, so 600 kts isn't a problem at FL300.
I step climbed to 30,000, after leaving Auckland, and checked the tailwind at around 72 kts. IAS was around 425 kts, so I increased altitude to FL370 and increased speed. At 35,000 ft, I started to notice a big drop in speed and checked the tailwind at 70 kts. At 37,000 ft. I was lucky to hold 350 kts with a tail wind of 67 kts. Other than that, I didn't get any unexpected loss of altitude, or anything like that. But there is something not quite right in that area. Beats the snot out of me. ???
Even though I don't usually fly that high, either, flying over California, Nevada, and the Western US, would normally put me around 550 kts at 37,000 ft.
I think PropStrike found a major bug for that part of the world.