I strongly believe the characters should keep their same names until we know for sure something changes. Think about it, how do we know if they'll ever address their full name again, characters have the habit of just saying first names and nicknames. So that's like waiting for something that's probably never going to happen. Arrow is almost over, do you think they'll call Lyla by her full name? If you say "Oh we can't assume their full-name" that's like saying we can't assume their SPECIES, so you might as well go back to Clark Kent's page and erase "Kryptonian" from his page but that would be ridiculous right?

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