Yes they stay open (1680x1050)
But I find it a bit odd that I don't see which one is actually active, so I usually click the buttons another time, starting from the left until I "find" the option I want. The whole thing has something of a try and error approach. It would help tremendously if the active one is higlighted. Perhaps that has to with the icons looking kind of random to me too ..
Another small issue, it can happen when I load another savegame, the faction menu on the right still has French units active, though I switched to Central powers.
Yet another one, when I move the mouse over the NATO counter it shows the elements on the left, but once there are > 4 units it creates a second row which interferes with the mini map. Perhaps it would be better single line. I would also find it nice if the old behaviour ould be back, that the pictures are shown on the left when the mouse is anywhere over the whole icon (not just NATO counter), and the number of troops shown for each unit when ALT is pressed.