Are you asking to get answers about the philosophical question, or did you argue with her about it and you want to find out what the lemmynet thinks?

As others have mentioned, “complete” does not have to refer to the puzzle itself. It could refer to the effort.

On the other hand, her puzzle, her frame, her wall: she gets to decide.

Personally, I’d want to craft replacement pieces and draw in the missing parts of the picture. Or maybe paint them gold like Kintsugi. That way the puzzle can be “complete” while still acknowledging the history of the puzzle.