First of all, it’s not a “love letter”, it’s a “tape”! It's the one picked up in the new building, the one used to trigger the plot after being turned to the front, and kept after use.
Secondly, the 4th floor of the new building is not a trash can, but a "complaint box".
Putting the used "tape" in the "complaint box" can greatly improve Shengya's favorability. The reason is mentioned in Shengya's dialogue. Also, you have to take the "Seung-ah vs. Ji-hyun" line to trigger the plot. Seung-ah's favor in the "Xiaoying line" and "Ji-min line" doesn't make much sense. After that, as long as you don't do anything weird and have smart conversations, you can trigger "Tuberose".
The "Love Letter" is used to trigger the collection of ghosts. After getting it, just check the handprints on the window of this building a few times (difficulty or above)