It's supposed to. I've always found it buggy and unreliable but maybe that's just me. (This command exists in Claude btw not sure about Codex)

Looks like codex has it too since last week, https://github.com/openai/codex/releases/tag/rust-v0.130.0

You can also connect remotely. Tailscale to connect to your network/machine. Then use SSH to login. Then use tmux to persist the session even if you log out.