[FEATURE] Desktop: session-generated "key info" pane with clickable entities that open in new Browser pane tabs

Problem

During a session I work with many external entities: PRs, pipeline runs, work items, deployments. Their links get buried in the transcript. Clicking one currently replaces the active Browser pane tab (#99692), and there is no compact place that keeps the key information of the session in view.

Proposal

Let Claude (or a plugin/mod) render a small, persistent "key info" web app pane next to the conversation. It would show the essentials of the session, for example:

PR 1234 [open in new pane tab] · Pipeline run 2345 [open in new pane tab]

Requirements:

1. A pane type that renders a small Claude-generated HTML app (similar to Artifacts, but local and bound to the session).
2. Entity links open in a new Browser pane tab (related: #84322, #99692).
3. Claude can update the pane during the session, for example after opening a PR or starting a pipeline.
4. Available in the desktop app. Mods can currently draw panes in the terminal, but not in the desktop app or VS Code (#99423).

Alternatives

• Pasting links in chat (buried in the transcript).
• A self-hosted dev-server app opened in the Browser pane (works, but is manual and not session-aware).

Related

#84322, #99692, #99423, #99502