ogglobi/lmarena.ai-chat-downloader ? reverse-engineered prompt

Build me a Tampermonkey userscript for LM Arena that lets me save chat conversations as a clean Markdown file.

I want two floating buttons to appear in the bottom right of the chat page. One button should automatically gather the whole conversation by scrolling through it and then download everything as a markdown file. The other should let me click messages one by one in the right order, like user then assistant, so I can fix the order if the automatic version misses something.

Please make the export easy to read, with the speaker names, timestamps if possible, and a simple markdown layout. It should work on lmarena.ai and chat.lmsys.org, and it should feel polished enough to use right away. If you need to check the current site behavior or docs online, go ahead and do that.