This page of the guide to Greedfall has the walkthrough for the of The Experiments of Dr. Asili's side quest.
Where to start the quest?: Complete the previous side quest At the center of the Rebellion.
Go to the lab, and it turns out that it is guarded. You can break through with your sword or just use the scaffolding nearby to go to the other side. Inside you will fight a swarm of enemies. Defeat everyone and make your way to Dr. Asili's quarters.
Downstairs you will find the doctor's assistants. You can order them to leave or stop them. In this walkthrough, we let them go. In this room, you will also find the keys to the cells - they will be useful a little later on.
Run upstairs and confront the doctor. Defeat him in combat, and then decide what to do next - you can either kill or apprehend him. In the case of the walkthrough, we chose the more civil option and apprehended the man.
Get out of the building and start freeing the prisoners using the keys you have found earlier. Then, go down into the gorge, defeat the guards and escape the area using an alternative passage. Notify Governor Burhan about the whole incident, which will complete the task - you will get + 2 reputation with Bridge Alliance. In addition, for releasing the Doctor's assistants and leaving him alive, you will get another +2 reputation with the Alliance Bridge. You finish the quest, you will get 1900 XP and start The Trial, another side quest.