Recorded on PS5, shared direct from console unedited. I tried for hours, differing steps, this order worked for me.

NOTE!!: This worked for me on PS5, it hasn't been working for everyone though. I wanted to share in case this works for anyone else. If you're on PC, check the pinned comment for a 100% working fix (thank you to @joe1690!!).

My PS5 Steps:
1) Logged in, saw there was an event in the area, i already have the quest to go into the dungeon/cave area but hadn't gone to speak with them yet.
2) Clear the event.
3) Once cleared, abandoned the quest.
4) Travel over to pick it back up again (used my horse)
5) Went into question location cave.
6) Spoke to NPC from the highest position i could (closest to gate)
7) Immediately ran over to gate after talking to her.
Note: Notice the gate DID NOT have a health bar at this point.

This is the first time i managed to get this quest to work, they ran up to initiate combat after this.

Leave me a like and a comment if it worked for you too. It'll let others know it worked for you.

Cheers!