... that the fixed stars are higher than previously suspected? Yes.
It's hard to observe satellites in high orbits, but showing that the stars are more then 3000 miles away is easy.
I know you don't think that the Earth orbits the sun, but the fact remains that the "fixed" stars
do shift slightly relative to each other throughout the year. (To the exact amount predicted by heliocentrism by the way) This effect is called stellar parallax, and it's how I got the idea to measure the altitude of the ISS in the first place.
Using this method, the Hubble telescope can accurately measure the distance of stars that are about 100 light years away.
Hubble is part of the conspiracy?
Ok, how about this. Good ground based observatories can measure stars distances up to about 20 light years away.
Professional astronomers are in on the conspiracy to?
Ok, fine. I personally can observe stellar parallax with my telescope for a few of our nearest stellar neighbors. They are not fixed in some "celestial sphere." And while I don't have a powerful enough telescope to measure their distances to the light year, I can tell you that they are much more then 3000 miles away.