Why was the fire/scarring angle not pursued in Memories of Murder?

While in real life, the investigation was not solved for a very long time, in the film Memories of Murder there was one conclusive lead that wasn't pursued.

When shown a picture of Park Hyeon-gyu, the restaurant owner's son (the one with the scarring) heavily implies that Hyeon-guy was the person who scarred him when he was a child. However, the police never pursue this angle, telling the kid to snap out of it. Why?