I had a car accident a month ago

I am 100% at fault,
and the accident happened on September 10th.
While making a right turn, the car in front suddenly stopped,
and I tapped them from behind—it was just a light bump.
Only the license plate was pushed in and dented.
The property damage was settled for about 1.2 million KRW.
They requested personal injury claims, so I opened claims for two people: the driver and their young daughter.
They haven't received any treatment yet,
so is there a reason why the case hasn't been closed?
I thought the case would be finalized if they don't receive treatment within a few days.