Remote Control ignores settings.json effortLevel (resets to medium on Desktop, high on mobile)

Related to (now locked) #51549.

I run a boot-persistent resident Claude Code agent on a homelab server via `settings.json` with `"effortLevel": "low"` set explicitly (via a jq patch in a bootstrap script). That works fine for CLI-originated sessions.

But when I connect to that same session via Remote Control from Claude Desktop, effort silently resets to `medium`, and from the Claude mobile app it resets to `high` — both ignoring the `settings.json` default entirely. If I forget to manually reset it after connecting, I end up sending queries at `high` effort that I explicitly configured to run at `low`, with no warning that the override happened.

Would really appreciate either:
1. Remote Control sessions inheriting the `effortLevel` from `settings.json` instead of resetting to a client-specific default, or
2. A non-interactive way to set effort (e.g. `/effort low` usable over Remote Control, not just the local interactive picker) so it can at least be re-applied on connect.

Re-upping this since it's a real footgun for anyone using settings.json to pin effort level on a shared/remote session.

Start with the settings.json effortLevel handling and the Remote Control connection entry point, comparing behavior with CLI-originated sessions. Use locked issue #51549 and the /effort interaction as context, then verify that Desktop and mobile connections preserve the configured value or provide a non-interactive way to reapply it.

Escrito por el modelo de indexación a partir del texto del issue.