Jack and the Cuckoo-Clock Heart

Miss Acacia finds Jack in front of Madeleine's grave, shivering in the cold. She tries to use the key to turn his clock but he throws away the key, choosing to die from not using the key. Then they finally kiss. When Jack breaks the kiss, time has stopped for him. Jack sings a song about climbing to Heaven while using the frozen snowflakes as a ladder, ascending into the sky as the film fades to black.

Question: does jack die in jack and the cuckoo clock heart
Answer: yes