If Dusk Rose Reliquary leaves the battlefield before its last ability resolves, the target permanent won't be exiled.

Auras attached to the exiled permanent will be put into their owners' graveyards. Any Equipment will become unattached and remain on the battlefield. Any counters on the exiled permanent will cease to exist. When the card returns to the battlefield, it will be a new object with no connection to the card that was exiled.

If a token is exiled this way, it will cease to exist and won't return to the battlefield.

If a double-faced card is exiled this way, it will return with its front face up, no matter which face was up when it left the battlefield.