The King Under the Mountain has agreed to your request to open the tomb of Thorin Oakenshield and ensure the Arkenstone remains within, but he wants you to first obtain permission from Thorin's sister Dís.
'You wish to open the tomb of Thorin Oakenshield and ensure the Arkenstone remains within? None could have breached the tomb without our knowledge, ${PLAYER}. And yet, even as I speak these words, I am gripped with uncertainty. If the Heart of the Mountain has been taken, it might explain the diminishment of our craft-skill....'
The King Under the Mountain strokes his beard thoughtfully.
'Yes, ${PLAYER}, we should learn if the Arkenstone still lies within the tomb or if some enemy has taken it. I alone cannot give you leave to open the tomb of Thorin Oakenshield, though I am King Under the Mountain and possess lordship of all that lies within. It would not be proper to open Mazal Akrâz without the permission of Thorin's sister, Dís. Her children also lie in that place, and I would not have their rest or that of her brother disturbed without her leave.
'She is the Keeper of Raven-hill, ${PLAYER}, and that is where you will find her. Speak to her and learn if she will allow the tomb to be opened.'
Gandalf wishes to speak with you by the throne of King Thorin III, in the Hall Under the Mountain.
Dís is at Raven-hill, south-west of the Lonely Mountain.
You offered to clear the brush and stones from the path up Raven-hill for Dís.
You volunteered to bring water from the well at the base of Raven-hill to Dís.
You volunteered to defeat foes in the vicinity of Dale for Dís.