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Background
You'll trigger this task when you search the trash container (#1) behind the
Whirling-in-Rags (#3) and find the victim's clothes inside.
Walkthrough
When you talk to Garte (#3) about the clothes, he'll deny putting them in the trash container, and he'll claim that nobody on his staff would tamper with a crime scene. He'll also mention that besides himself, only the Trash Collection Service has the key to the trash container. If you ask Cuno (right next to the trash container) about the clothes, then he'll also deny knowing anything about them.
Starting on Day 3, you'll be able to head into
Martinaise West, where you'll meet Gary, the Cryptofascist (#4). He'll call Kim a "Yellow Man." If you have the
Yellow Man Mug from the trash container, then you'll be able to ask him if it belongs to him. After some dithering, he'll start admitting everything: that the mug is his; that he knows someone from the Trash Collection Service, who gave him a key to the trash container; and that he was the one who threw away the hanged man's clothes.
During Gary's admissions, if you pass a passive Perception check, then you'll hear a clinking sound when he moves. If you ask him about it, then he'll deny that it's him. He'll also deny knowing anything about the hanged man's armor. To get Gary to fess up, you'll need to pass a "legendary" Composure check. You can get some help with this check if you recognize the clinking sound from the hanged man's boots, if you get Morell (right next to Gary) to admit that Gary has been acting strangely lately, or if you've been inside Gary's apartment (#2), which you'll unlock during the task
Open Apartment Door for Evrart.
When you pass the Composure check, Gary will cave in again, and he'll quickly give you the
Fairweather T-500 Cuirass. This will complete the task and earn you 30 xp.
Note: After starting the task
Inspect the Traps, Gary will move into the
Whirling-in-Rags (#3), and you'll still be able to talk to him there. But after completing the task, Gary will return to his apartment, and he'll disappear from the game. So if you want to complete this task, then you'll need to do it before completing
Inspect the Traps.
1 - Trash Container
2 - Gary's Apartment
3 - Garte / Whirling-in-Rags
4 - Gary