Firefox does this, but the tabs remain private, because it leaves no data from that session after the windows are closed.

Why don't private windows allow you to reopen closed tabs and get search suggestions? It's such a pain in the ass when you accidentally close private tabs and don't remember exactly how you got to that page.

As for the search suggestions, I'm aware that there are people concerned about sending data to the search providers, so it could be made optional.

What do you think?