Claude in Chrome: name tab groups after the session instead of generic "Claude"

Feature request

When Claude Code connects to Chrome via the Claude in Chrome extension, each session's tabs are placed in a tab group labeled "Claude". With several concurrent sessions, the tab strip fills with visually identical "Claude" chips — in my case 7+ groups named "Claude" at once — and there is no way to tell which group belongs to which session.

Request: name the tab group after the Claude Code session (the session name shown in the CLI / set by the user), or at least expose a per-session label. Even a truncated session name would make the groups instantly distinguishable.

Current behavior

• Every session creates a new tab group with the fixed label "Claude" (color varies).
• Manual renaming works per-group, but the next session spawns another generic "Claude" group, so it doesn't scale.

Prior art

• #17536 ("Renamable/Named Sessions in Claude Code Extension") was closed as not planned — asking to reconsider, since multi-session use has become the normal workflow (parallel worktrees, task-board setups).
• Related pain: #23861 (MCP tab groups persist and multiply across sessions), #25284 (setting to disable tab group creation).

Environment

• Claude Code CLI on macOS (Darwin 25.5), Chrome with Claude in Chrome extension
• Many parallel sessions against the same repo (worktrees + shared task board)