Feature: Isolate browser sessions per task or project

Problem

The in-app browser appears to share cookies and other website session data across Codex tasks. This prevents parallel tasks from using different accounts on the same service.

Example:

1. Task A signs in to example.com with Account A.
2. Task B signs in to the same site with Account B.
3. Returning to Task A shows Account B's session.

This is inconvenient and can also cause actions to be performed in the wrong customer, organization, test, or production account.

Requested feature

Add selectable browser-session isolation:

• shared, matching the current behavior
• isolated per project
• isolated per task

An isolated context should have its own cookies, local storage, and authentication state. The selected mode should be visible in the UI.

Expected behavior

Task A can remain signed in as Account A while Task B uses Account B on the same website.

Related issues

• #39552 notes that in-app browser tabs share one persisted storage partition.
• #23314 requests multiple browser tabs, which does not provide cookie isolation.
• #21145 requested external browser profiles; this request is specifically about isolation inside the in-app browser.

Feedback requested

Is this only my workflow, or would others also benefit from per-task or per-project browser sessions? I would especially be interested in use cases involving multiple customers, organizations, environments, or user roles.

Read related issues #39552, #23314, and #21145 first to understand existing browser storage, tabs, and profile constraints. Then locate the in-app browser's session/storage and mode-selection entry points; done means shared, per-project, and per-task modes isolate cookies, local storage, and authentication state while making the selection visible and supporting the Account A/Account B scenario.