Here's an idea. Not tested, but feel free if you have any other ones.
When you decide to run: SKStoreReviewController.requestReview() so the user is prompted with the native "Rate in the app store",
Set and control a timer.
Then in the appdelegate/applicationWillEnterForeground control this timer:
- If the user selected "submit", the app will probably go to background within the next 5 to 10 seconds. So if the app hit this method and the elapsed time is less than (i.e) 10 seconds, than (probably) means that the user went to the Apple Store.
- Meanwhile, if the timer hits 30 seconds, stop the timer and save your guess: The user didn't submit the review.
Also, many apps present a custom popup asking the user if they like the app and if so then they present the native SKStoreReviewController.requestReview()