In Conquest of Planet Earth, Do I conquer a Location with my remaining Alien Ships after Losing the final Fight Round?
I don't think the line "you may start a new Fight Round" means that the Alien Ship can stay on the Location, or claim a victory.  I think it means "you may carry on trying to defeat the Resistance by starting a new Fight Round... or choose another valid option".

What other valid options are there?  See the next section of the rules:


  After each Fight Round is resolved, the Aliens have the opportunity to
  escape; immediately ending the Battle as though the Aliens had lost.


I think it's completely intuitive, based on other similar boardgames, that you should have to keep attacking until you either win fairly or are forced to flee.  No sneaky short cuts!