Recently, I scored a good deal on a 2nd hand French press. This is my first time owning one, and this thing was last used god knows when. Couldn't get the stains out of the common mesh part, so that's likely getting replaced. What I'm not sure about is this additional fine mesh filter (kinda feels like a reusable coffee machine filter) that I'm not seeing in any French press cleaning videos on youtube. Dish soap isn't getting this clean at all. Do I need this thing or can I chuck it? The other parts seem straightforward enough to clean, but if there are any useful tips for getting this whole press sterilized, I'd really appreciate it. Would a diluted vinegar bath help for example?

https://preview.redd.it/ay6b4xf32csd1.jpg?width=1536&format=pjpg&auto=webp&s=927dffb4914ec14e5c639ee59edbf85e6a201010