How to get this quest
After completing Festival of Greed, talk to the white priestess in Trademond.
Reaching Mountainreach
The first step of the quest is to reach Mountainreach. For this, you'll have to cross the bridge leading to Lumeria. The bridge is guarded by bandits. Two options can deal with them:
If you choose to fight, note that you'll get a mini boss battle battle as the final fight (the toll bandit leader accompanied by multiple underlings). All fights are consecutive. Losing leads to end the day or take the above lewd option. Ending the day doesn't reset the number of fights you won. You can go to the bridge again to finish the last fight. Also note there is a bounty on the toll bandit leader and Julian can reward you for his defeat.
The best way to quickly win all fights is to begin with Shadowcloak on and a fire slime summoned. With enough ATK, using Slash on the first turn will dispatch the toll bandits. The city thugs will get crippled and easy to finish. The leader becomes easy when he is alone.
Missable items
There is a shop in Mountainreach where an alchemist named Louise sells some items. Among them is the formula for Aphrodisiac Bomb Drug+, which can be given either to Nephlune's white priestess or to the underground drug dealer. This shop won't be available after completing the quest.
Learning Open Domain II
After arriving in Mountainreach, enter the church and talk to Elaine the exorcist. The spell Open Domain II will be available the next day to learn. You need 100 MP to learn it.
You are a point in the game where demons are a source to raise your max MP (Drakhnor Fortress is a good spot) and the northern mountain ranges have some mana stones behing the goblin king. Of course, mutated spores and mana stones from the alchemist are also an option (ether is even buyable in Nephlune now).
Reaching Belphegor
Once you end a day inside Mountainreach, you'll get a partial (1/10) Legs of Sloth curse. From now on, whenever you end the day in Mounainreach, the curse will gain 1/10 progress. Once the curse gets completed, Belphegor will escape.
Belphegor's domain has multiple entry points. Those points are Mountainreach's inhabitants. You can confront Belphegor through Sophie. Once it is done, two possibilities occur:
To purge all negative energy points, Aaron need to get back on his feet. The first time you enter Mountainreach, he is bedridden in his house. You can manipulate the time loop to erase his accident. For this, go to Louise's shop and raise her bomb stock to 10. The next day, Aaron will be working outside. You can use Open Domain II on him to access a negative energy point which would have been unreachable otherwise.
Fighting Belphegor
Once you get past Belphegor's barrier, you have to fight him accompanied by a reconstitution of Luciela.
Assuming you want to defeat Belphegor without using the Star Knightess power, here are some advice:
The day ends whether or not you win the fight.
End of the quest
Get your reward from the white priestess. Some other rewards can also be obtained: