he asked 2 pi rotations, not 1 rotation
I did not say 2 pi rotations, I said "a 2 pi rotation" Specifying one revolution as measured in radians.
You can observe stars in the sky returning to approximately the same point every 24 hours, suggesting that if your earth is spinning within the star hole sphere you suggest, the rotation rate is 1 revolution per 24 hours.
I also, however notice a new problem: That the stars on the sphere would appear to grow brighter and further apart as they approach the horizon, since they are getting closer. Also since the earth's axis of rotation runs parallel to the surface and above the surface of the earth, you should be able to see, from anywhere on the earth, two points above the horizon around which the stars appear to revolve. This is not observed. If you find time lapse videos or long exposures of the night sky, they always show the stars revolving around one point of revolution above the horizon, with the other point of revolution hidden below the horizon, or both points of revolution right on the horizon.