What is the hero's quest in The Buried Giant?

In The Buried Giant, the two main characters (our heroes), Axl and Beatrice, live in a landscape over which a mist has fallen during a period after a war between the Saxons and Britons. The mist is caused by the breath of a dragon who lives in the hills. This mist seems to cause mass forgetfulness and changes memories of bad events to seem to be scary visions.
Axl and Beatrice, although they hardly remember him or why he disappeared, set out to find their lost son. Their quest, which seems at first to be locating their son, becomes a quest to defeat the mist and slay the dragon.
Once the dragon is slain and the mist is gone, people will get their memories back, good and bad, and recover their pasts. However, the period of forgetfulness also created a period of peace between the Saxons and Britons, who’d been able to forget their hatred. With the memory of the land recovered, the possibility of war and bloodshed is great.
A hero’s quest typically involves a journey which ends in victory. The couple does succeed in vanquishing the dragon, but the idea that this is a total victory can be debated. If it isn’t a victory, does this journey count as a “hero’s quest”?