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Jaewoo is still the same person, just the hair is flattened and has less volume. I don't get why people are so affected by it, i mean he did got transferred to a novel as a DIFFERENT character yet not much has changed to his body. Read another damn manhwa cause it's obvious that you are NOT getting the plot here ヽ(`Д´)ノ. Anyway, Jaewoo should've told the truth but i guess jaewoo telling the truth would be too silly for him to actually believe that.
I feel like Mujin will definitely think that it's Jaewoo, there isn't much difference from the old jaewoo, they'd be too blind to not see