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Am I correct then thinking that if I put 13.1 beta on my iPad Pro, and then want to upgrade to iPadOS 13.1 GM on September 30th, it won't work? I've been holding off with the 13.1 betas because of this.

This will work and it's what I've done in the past. You just need to remove the beta profile from your phone... say on the 29th and then on the 30th your phone will update to the general release of 13.1. Apple's documentation states this as well...

How do I stop receiving public betas and install the next shipping iOS, iPadOS, macOS, or tvOS release?
When your device is enrolled in the Apple Beta Software Program, you will automatically receive new versions of the public beta from iOS Software Update, the Mac App Store, or tvOS Software Update. At any time, you may unenroll your device so that it no longer receives these updates. Then, when the next version of commercial software is released, you can install it from iOS Software Update, the Mac App Store, or tvOS Software Update.
 
This will work and it's what I've done in the past. You just need to remove the beta profile from your phone... say on the 29th and then on the 30th your phone will update to the general release of 13.1. Apple's documentation states this as well...

How do I stop receiving public betas and install the next shipping iOS, iPadOS, macOS, or tvOS release?
When your device is enrolled in the Apple Beta Software Program, you will automatically receive new versions of the public beta from iOS Software Update, the Mac App Store, or tvOS Software Update. At any time, you may unenroll your device so that it no longer receives these updates. Then, when the next version of commercial software is released, you can install it from iOS Software Update, the Mac App Store, or tvOS Software Update.
Oh, at the time I meant iOS 13.0 GM. But I've since learned that it should be 13.1 on the 30th. Funny that I made a typo that was actually what was happening and didn't even realize it. Apparently I didn't notice it at the bottom of the article. Info was flying quickly after the event.
 
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