I am running iOS 9.1 beta and it seems pretty good and more stable than past iOS 9 iterations.
Will my iPhone 5S prompt me to upgrade to the public release of iOS 9 today?
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You will not be prompted to update today since the 9.0 release will be lower than the 9.1 beta you are running. You could downgrade, but that's really dependent upon what your specific needs are going forward.