[BUG] Remote Control session shows/processes incoming cross-session messages but reports 'offline' for the user's own input

On my phone, using Remote Control to view one of my Claude sessions:

**I cannot type a message into the session. When I try to send, I get this exact banner:**

> Remote control session is offline. Reconnect or run /remote-control to start a new session.

**But at the same time, messages that ANOTHER Claude session sends to that same session DO show up, get answered, and I watch the task run live on my phone.**

So the session is clearly not offline. Another session can reach it and make it respond, and I can see all of that happen on my screen — but my own typing into the same session, on the same screen, at the same moment, is blocked as "offline."

Simple version: I can't write to it, but I can see what another session makes it write.

**And the word "offline" no longer means anything here.** You are telling *me* the session is offline, while I am watching it receive messages, run tasks, and print answers in real time. If it were offline, none of that could happen. So "offline" in this banner does not describe the real state — the word has lost its meaning in this bug.

Steps:
1. Open a session on the phone via Remote Control.
2. Have another Claude session send it a cross-session message. → It appears and the session starts working on it, live, on your phone.
3. Now type your own message into that same session and send. → "Remote control session is offline."

Both things happen at the same time on the same session. That's the bug.

Environment: Claude mobile app, Remote Control to Claude Code sessions on a Windows desktop.

No source file or test is named. Start by reproducing the multi-session Remote Control setup and trace how session connectivity is determined for phone input, live updates, and the SendMessage cross-session tool. Done means these surfaces consistently reflect the session's actual ability to receive and process messages.

Written by the indexing model from the issue text.