Yes, they are reincarnated. From their previous life wherein Hosik died because of an illness, Hyung kept on dreaming about that. Meaning, he remember their past life but Hosik don't. They are in their second life.
It just one of their past lives, to explain it better: the 1st timeline was the one Hosik died, we could count this the 2nd timeline where Hyung is desperately looking for Hosik (bc in the 1st one he died leaving Hyung all alone and losing his most precious person) and finally the 3rd one that is what we read we them living happily as family. I think is a little confusing at first but if you reread bc the story is really good if you count all that they suffer life after life just to be together.
I am so confused, like did they die and reincarnate, or is it just a non canon reincarnation side story. Please someone explain