Brief Question for D&D Players Re. Cloakwood/Candlekeep/Gullykin

[The_Baffled_King]

Gullykin

Is Gullykin under the jurisdiction of the Grand Dukes, the jurisdiction of Amn, or neither of the two? If the answer is "neither of the two", is it under the jurisdiction of another political entity, or is it essentially an insignificant village in the middle of nowhere that is left to do its own thing?

Candlekeep

To what extent is Candlekeep under the jurisdiction of the Grand Dukes, if at all? If it is not under their jurisdiction, then what would it take - both in terms of the trigger required to act, and the force required - for the Grand Dukes to enforce their will on Candlekeep (assuming of course that it is even feasible for the Grand Dukes to do so)?

By "enforce their will", I mean "credibly threaten Ulraunt with consequences serious enough to force him to do something he otherwise would not do" - said consequences do not necessarily have to be so serious as laying siege to Candlekeep (it depends on what Ulraunt was being asked to do), but I'm curious about whether the Grand Dukes could credibly threaten to take control of Candlekeep if they were sufficiently motivated to do so.

Cloakwood

To what extent is the Cloakwood under the jurisdiction of the Grand Dukes, if at all? As a related question, is there any political framework through which the remaining Orothiar dwarves (such as Yeslick) should have been able to enforce a complaint that the Iron Throne had stolen their property (the Cloakwood mine)?

For me, this is the question that I'm most interested in. I get that the Iron Throne would want to keep their Cloakwood mine a secret while the iron crisis was ongoing given that they were using slave labour, and they later began to use slave labour in the form of people abducted by the Chill and Blacktalons.

But I'm wondering what the position should be according to D&D lore once the existence of the mine is known in the city of Baldur's Gate - especially with Yeslick freed.