Group sessions by project in the mobile and web Code session list

### Preflight Checklist

- [x] I have searched [existing requests](https://github.com/anthropics/claude-code/issues?q=is%3Aissue%20label%3Aenhancement) and this feature hasn't been requested yet
- [x] This is a single feature request (not multiple features)

### Problem Statement

I run six separate projects with Claude Code Desktop on one Windows machine and steer them from my phone through Remote Control.

On the desktop, the sidebar groups sessions by project folder with a collapsible header per project, so it is immediately obvious which session belongs to which project.

The mobile app and claude.ai/code show a single flat list ordered by recency. There is no project grouping, no project filter, and no indication of the working directory. The filter menu only offers session state: All / Working / Needs input / Ready for review / Completed / Archived.

With five or six connected sessions the list becomes unreadable. Titles are truncated on a phone screen, so the project name is often the part that gets cut off. The only way to tell sessions apart is to remember to prefix every session title with the project name by hand.

Remote Control is most valuable to people running several projects in parallel and checking in from a phone. That is exactly the audience most likely to have many sessions at once, and the flat list is where the feature stops scaling.

### Proposed Solution

Mirror the desktop sidebar behaviour on the remote surfaces:

1. Group Remote Control sessions by their working directory / project in the mobile app and on claude.ai/code, with collapsible headers, the way the Desktop sidebar already does.

2. If full grouping is too much, a smaller version would already help a lot: show the project (folder name) as a subtitle on each session card, and add a project filter alongside the existing state filters.

The desktop app already knows the folder for every session, so the grouping key exists. It is simply not surfaced on the remote surfaces, where the second line shows "Remote control" or a GitHub repo name instead.

### Alternative Solutions

Manually running /rename in every new session with a project prefix, for example "/rename ProjectA - fix the import". This has to be repeated for every new chat and is easy to forget. Titles auto-generated from the first prompt do not reliably carry the project name.

Naming the project in the first message of each chat, hoping the generated title keeps it. Unreliable, because the title is a paraphrase of the prompt, not a copy.

Using the state filters (Working / Needs input) to narrow the list. Helps with "where am I needed now", but says nothing about which project a session belongs to.

### Priority

Low - Nice to have

### Feature Category

CLI commands and flags

### Use Case Example

_No response_

### Additional Context

Related but different: #60954 and #38036 both ask for project grouping in the Claude Code Desktop sidebar, which now works. This request is about the remote surfaces — the Claude mobile app and claude.ai/code — where the session list is still flat and shows no project at all. Asking for parity with the desktop sidebar on the surfaces you use when away from the machine.

[Документ Microsoft Word.docx](https://github.com/user-attachments/files/31505212/Microsoft.Word.docx)

No files or tests are named. Start by comparing the existing Desktop sidebar grouping with the mobile app and claude.ai/code Remote Control session list, then find where session cards and filters are rendered and whether working directory/project metadata is available there. Done means remote sessions are grouped by project or show project subtitles plus a project filter as described.

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