I think so too, but visually, the parent switch in the macOS application is Chat GPT <-> Codex, and within ChatGPT, there is Chat <-> Work (and within Work, there is "on your computer" and "in the cloud"). I'd naively not expect ChatGPT -> Work to share state with Codex, but this is the case for "on your computer" tasks.

Also, there's still "Codex Cloud", much older and entirely separate from all of this (I think it might even predate Codex the local app?), and this also seems to live in the "Codex" parent tab on a project by project basis.

None of this seems to be documented either. I literally had to start a "Hello World" project in each category to figure out what's going on.