Just for fun, I opened the observation window. Should I close it, or can I leave it open during June?
I would be inclined to keep it closed. You will find as I do that you'll have bees constantly trying to exit there. By keeping it closed, they will only try to use the entrance, with success.
do you mean the window on the flow super? Several problems with leaving that off- apparently the plastic of the flow frames will degrade faster if exposed to sunlight. The next issue is the wood - the window cover may change size relative to the box if they are not in the same conditions- and you may find it hard to put the cover back on if it expands and/or the box shrinks. I accidentally left mine off overnight- and the next day it would not fit back in. I had to put it in the sun for a few days before it shrunk back down enough...
Totally agree with @JeffH. Close it. About 50% of the hive is averse to light (nurse bees, queen etc.), so you are stressing the hive if you let light into it.