From the screenshot you can see that system icon (1) says that Cursor have one unread completed job, but actually (from the chat lists (2)) you can see that I already read all.
When job is done and I read it, the state at system icons should be flushed as it was flushed for the chat (you do not see blue circle on the right of this chat).
Thank you for the report, @Eugen_Konkov! I was able to reproduce this issue and have linked it to an existing bug ticket we’re already tracking. No ETA, but reports like this help us to prioritize!