Hi there! It seems like you're facing an issue with the Firefox icon disappearing from your taskbar after closing the browser. Don't worry; we're here to help! It's possible that there might be a simple fix for this. Please try the following steps:

Right-click on the taskbar and select "Taskbar settings." Scroll down to the "Notification area" section and click on "Select which icons appear on the taskbar." Look for Firefox in the list and make sure it's set to "On." If it's already on, try turning it off and then back on again. Restart your computer to see if the Firefox icon stays on the taskbar after closing the browser. If the issue persists, feel free to come back and let us know. We'll be happy to assist you further to get this resolved.

Thank you, but this doesn't solve the problem. I am logged in as admin and FF is the default browser. I pin the FF icon to the taskbar, open and close FF, and it disappears from the taskbar. I then follow your guidance, but (a) FF is not even listed as an available taskbar icon in the notification area of taskbar settings, and (b) even when I opt for all icons to display in the taskbar (which gives me HP Audio, Windows Security and a few other extra icons that are normally hidden), opening and closing FF makes the FF taskbar icon disappear again. Is it perhaps significant that this is only happening on my new HP laptop (Win 11 capable, but still running Win 10 Home), not on my older Win 10 Pro laptop that won't run Win 11?