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There's no reason at all for it to slow down. If you've ever been on one of those moving walkways think about it like that, yes by walking on it you're moving faster than the people around you, but to you you're not walking any faster than normal even though you'll arrive at your destination earlier.
That's actually a really good analogy. Mind if I throw some numbers on it to flesh things out?
Let's set that moving walkway going at 5mph, and we'll put ourselves on that walkway, on a turned-off rascal scooter. The scooter is stationary on the belt, but it's still moving at 5mph - that's your tailwind pushing the air around the plane forward.
Now, let's turn that scooter on and throttle it up to 5mph. The scooter is plugging along comfortably at 5mph, but it's actually moving at 10mph. This is your plane flying with a tailwind, performing normally for its indicated air speed, while having a much higher ground speed.
Curiously, this does make the phrase "supersonic speeds" somewhat debatable. While they were traveling over the ground faster than sound would, they weren't moving faster than sound would in the air around them.