> and slides it back and forth as the tab panel expands and goes back.

There's a "pin" button there that keeps it open. I know, this doesn't fix your problem, but I and everyone I've seen using it uses the pin button, and IMO it's a much better experience :) I've noticed that on most high-end laptops websites have way too much blank space on the side margins so pinning a useful tab bar in that space works fine.