Is there a way to add a custom favicon to an App Tab?

In the above thread, @admintech links to a great plugin for changing favicons which covers both the bookmarks folder and the address bar/tab bar icons. However, it still does not quite fully address what I was hoping to accomplish.

I'd like to set an App Tab that has a customized icon, that stays the same in that tab no matter what I do there. Since the navigation within an App Tab is very restricted, the chosen favicon should always be relevant to whatever page is loaded in that tab.

The Bookmark Favicon Changer has been effective in allowing me to use a custom favicon in the App Tab. But, the favicon only applies to the specific URL that was bookmarked. Any navigation done from that page will return the favicon to blank.

Is there another plugin, or perhaps some special tweak to this plugin or the bookmark itself, that will allow me to make the favicon more persistent across the site?