Apologies for not sharing an explanation on this earlier. This is actually a limitation on the iOS platform as they do not allow apps to go in the background. When the screen is being captured and streamed to the supporter device, the TeamViewer QuickSupport app needs to be backgrounded, and iOS will end up killing it as it has a short timer on apps backgrounded. Hence we stop it. So nothing is wrong from your side, this is how the feature is supported with the current platform limitations. Hope this helps.
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