Given that you're considering creating a Glitch-only browser, it might be worth some effort to simply remove the primary reasons that people have other browser tabs open.
If you didn't have to have another tab open to futz with skills or auctions, it would be much easier to contain everything within one tab.
I realize that'd be a sizable project and I'm almost certain it's occurred to you before, but, hey, you've got an Ideas forum so I'm gonna use it.