Capacitor video player, PiP button does not show/work on iOS

Picture-in-Picture does not work on iOS.
The PiP button is never rendered in the native player controls — it is not greyed out or inactive, it is simply absent.
Video playback itself works normally.

The same code on Android shows the PiP button and PiP works as expected.

Additionally, background audio does not work on iOS: with bkmodeEnabled at its documented default of true and UIBackgroundModes → audio declared in Info.plist, locking the screen during playback stops the sound.
It should keep playing. Both issues reproduce on every attempt on a real device.