Blighted Battleaxe, if I don't conquer the gear dies right?

No — the gear doesn't die. It unattaches and deals 4 damage to the unit, then gets Recalled back to your base.

Here's the sequence:

1. End of your turn — if the equipped unit didn't conquer, the trigger fires. "Me" refers to the equipped unit (FAQ 9662). The Battleaxe unattaches and deals 4 damage to that unit.
2. Damage heals — during the same turn's Expiration Step, Ending Special Cleanup occurs, which includes "Heal all Units" (Rule 317.2.b). That 4 damage is cleared before your turn ends (FAQ 11381).
3. Gear recalled — once unattached, the Battleaxe is an unattached gear sitting at a battlefield. During that same cleanup, it gets Recalled to your base (Rule 452.1 / Rule 149.3).

So the damage to the unit is temporary, and the gear survives — it just pops off and goes back to your base, ready to be equipped to something else next turn.