Does Blighted Battleaxe unattach damage persist to opponent's turn?

No, the damage from Blighted Battleaxe does not persist to the opponent's turn.

Here's the sequence: Blighted Battleaxe's trigger resolves during the Ending Step of your turn ("at the end of your turn"), unattaching and dealing 4 damage to the equipped unit. Then, during the Expiration Step of that same turn, the Ending Special Cleanup occurs — which includes the instruction "Heal all Units" (Rule 317.2.b). That healing clears all damage from all units before your turn ends.

So the 4 damage is wiped clean during your own turn's cleanup, and the opponent starts their turn with no remaining damage on that unit.