Can individual documents "remember" their Inspector tab?

There isn’t a setting like that. We hope to make it easier to switch between panes though in the future, perhaps with some keyboard shortcuts. I switch panes constantly, and I’ve never really noticed there being much of a burden in doing so—but I’m on a Mac. If I had to reach for the mouse and click a tiny little button in the corner of the screen every time, I’d probably get fed up with it.