I just did the "shield and survive" phase. All the enemies are dead and the ships are gone, but the boss still has the overshield and it still says "shield and survive". All the enemies are dead and ships are gone. The only enemy left in the room is the immune boss.
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I died and restarted. The next time through, the boss ran into the bubble and popped it. So I guess if this happens to you, then just die and restart.