I do not own any rights to Naruto.
Blame a morbid brain and the song 'He Stopped Loving Her Today'. Almost definitely non-canon.
In the end, they gave up on Sasuke.
It wasn't when he left; it wasn't when they found him again, only to lose him, either.
It wasn't through the long years of warfare with Orochimaru, through the deaths of many of their peers; loyal ninjas all, falling for their village. Autumn came to the village of Leaves, and it took too many away.
It wasn't when the loudmouth demon-brat became a hero. It wasn't when the hero calmed down, settled into the title of Hokage like it was made for him. Naruto never did settle into sending people out to die, but he never did if he could help it. He went out himself first. He faced Sasuke many times, so that others wouldn't die. One day he didn't come back.
It wasn't when Sakura got married. It wasn't when she had children. It wasn't even when she had grandchildren. All through her life she never gave up on a promise made in haste and never forgotten. She spent her life in hospitals, mending everyone but herself, so the stain on his soul would be as small as possible. One day she broke herself past repair.
In the end they gave up on Sasuke, because death breaks all promises.