There are not so many runs under 6k turns so I decided to investigate the lower bound for turns a bit. This is not a thoroughly optimized effort but more of a proof-of-concept run.
So, here's the character.
He was born in Cup under a falling star. 20% fewer experience to advance. Cup seems to negatively affect starting spell knowledge.
He was a gnome. Another -25% to the experience requirement.
He was a wizard. Wizards get reduced cost of spells, more spellbooks, level and learn spells easily, get spell knowledge starting at level 25, ability to recharge wands etc.
1. After entering the Drakalor Chain he headed to the ToEF. He made a pit stop at R4's pool.
2. Then he went on to deal with the ACW of the ToEF. The wyrm tried to run away but its endeavour was fruitless.
3. The wizard travelled to the Caverns of Chaos. He ignored temptations of the Arena and took no part in consuming lizards for gluttony or mortal combat for greed and entertainment.
4. Upon his arrival to Dwarftown he carefully removed an old bearded obstacle from a lever opening access to the lower levels of the caverns.
5. Using an earlier obtained amulet he saved Khelavaster.
6. He did not disturb dwarven graveyards.
7. After repeated attempts he managed to provoke the Snake of Water Temple to abandon its nest. The snake did not survive for long.
8. He critically hit Banshee saving himself from the approaching danger.
9. He teleported and hurried downstairs leaving the angry eternal guardian behind.
10. He did not engage in trade or gambling inside the casino.
11. He decided not to bother a man who apparently was a farmer breeding rabbits but could not help picking a ring dropped by a remarkable huge feline.
12. Whether the illusionist of the Air Temple was named after a speedrunner or not he was fired from that job.
13. A strange surge of power caused an artifact to appear (chain mail of the martyred crusader which was sacrificed later).
14. Just a level below the wizard stumbled upon four artifacts in a greater vault: Shezestriakis, Protector, Soaker and The Wall.
15. He defeated the stone beast of the Earth Temple using crossbow bolts and powerful spells.
16. He found a nice rune-covered long sword in a greater vault located 44 levels below ground (Trusted One). He also found a black book, a piece of map and heaps of gold.
17. His deity crowned him as a champion of law.
18. Deep inside the sidebranch of the caverns he saw a poacher inside his hut surrounded by many illegal traps and put an end to this.
19. He closed the chaos gates and destroyed the means to open them again.
20. He quickly teleported through the Caverns of Chaos to Darkforge and after a short stop in a small village left the Drakalor Chain.
Here's the flg file and the recording of all pressed keys with comments.