Woah, holy cow! There are about a hundred posts over three years saying it's a bug, impossible, close all windows, next release, make macros, jump up and down, fiddle with IE preferences, etcetera. This is the only post anywhere on EARTH that I could find where someone said check this, that, and it works. And it works!
Now if I have accesspoint in one tab, something else in another tab, close the access point tab, and reopen the tab it doesn't require the sign-in, but it did that anyway. It seems like the Authenticate->Always is failing there and the Authenticate->Login is actually stronger even though it's not an 'always'. If I sign out, I have to sign back in. If I close all browser windows and reopen, I have to sign back in. Sign out button is so much nicer than close all windows.