Question about Player Progression in ACKS

So I'm still GMing my first ACKS Campaign and is being a blast! I currently have 7 players and every one of them is loving the system!

I have a question about the progression that makes the central concept of the game, of players evolving from Adventurers to Conquerors and then to Kings. They are roughly level 3 and we are going to our 15th session. At which level they become Conquerors? At level 9? And Kings at level 14? I know there is no need for some hard rule stablishing that, and I like how things develop through RP, but I would like to know ate which point in average you guys usually make the transition from one style of game (adventurer) to the next (conqueror) and then to the last (king).

So I welcome any input from your experiences!

If anyone wants to know my players are:
-Hiwatta, a young Shaman 3 of 15 years of old, followed by his animal companion the wolf Tanuk.
-Adewale, a incredibly smart Barbarian 2, followed by his pack of hunters (3 henchmen barbarian 1)
-Habu, the ebony headhunter Fighter 3 (he previously played with Luhan, a Thief 3, but died twice and took control of Habu, his Fighter Henchmen)
-Dardok, the nordic sellsword Fighter 3
-Farid, the Mage 3, followed by his companion Nyb, a Thief 1
-Amma, a female Mage 2, followed by her younger brother Ali, a Fighter 1
-Silver-Feathers, a Priestess 1, newest adition to our group

All of them human. They just recently uncovered a plot inside the Church of Kyrios, the Last Living God, freed the slaves taken by the Sun Templars and are now deciding if they will hunt/pursuit the survivors who are trying to reach and warn the Archpriest of the region, or flee beyond the borderlands, to the Apokalypse Lands, a region without law, where monsters and beastmen reign.

Not an Autarch but IMO: Roughly speaking, Adventurer is levels 1-4, Conqueror is levels 5-9, and King is levels 10-14.

(edit:  ACKs Core page 253, in the starting characters above 1st level, says that Adventurer characters start with 20k xp, Conqurerors with 310k, and Kings with 620k.)

You can generally expect that adventurers will want to delve dungeons, conquerors will want to explore the wilderness, and kings will want to rule domains.  (Though of course these are expectations and there'll be plenty of variance at individual times.)

Given that ACKS is very much designed for sandbox play, this is something your players will probably decide for themselves: Sooner or later, they'll decide to diversify and do something different to what they've been doing so far.

For my players, it happened at around level three: While on the way to a dungeon they'd been exploring, a random encounter roll had them encounter an ancient tower covered in statuary and guarded by three gargoyles shaped like winged demons. Though careful strategy (and a number of lucky rolls) they shattered the gargoyles and explored their tower, where they discovered a magic portal that led to an identical tower in a vast desert.

My players (being enterprising sorts) promptly realised that this magical shortcut across the world had tremendous potential as a trade route, and decided to monopolize it by building a stronghold around it. The tower was in a borderlands area, and they had a fair amount of unspent treasure saved up, so they had little difficulty in doing this.

They've yet to successfuly find any markets on the other side of the portal, but they're working on it... In fact, the quest for a market on the desert side is the main motivation they have for venturing into the wilderness..