I am not sure about how it should work.
But here are some things I noticed.

quit and next should be local in LV2 and LV3 - change to self.quit and self.next

QApplication does not have reject(). I changed it to quit() in LV1

And finally, self.connect for quit in LV1 is defined inside nextWindow. Should be moved up before nextWindow.

I have attached the code. Changed geometry settings to windows don't overlap - easier to see.multiwin.py