Open the installed PWA from the home screen. A blue "PWA window — loaded at …" bar appears. Note the time.
Switch to this page in the Chrome browser (no blue bar here) and press L.
Where does the popup open: a new PWA window, or does it replace the PWA page you just opened (popup heading instead of the blue bar)?
Press Send message. Afterwards, open the PWA again from the app switcher: is the blue bar still showing the same "loaded at" time?
Experiments
Each button opens a popup. In the popup, press Send message. The result shows which
channels delivered the message and whether this page could close the popup.