[Feature Request] Claude Desktop Cowork: Support Microsoft Edge as browser connector (not just Chrome)

Feature Description

Claude Desktop's Cowork mode (browser connector) currently only supports Google Chrome. When using Cowork features like "Get tabs", Claude Desktop shows the dialog "Check your Claude in Chrome account" and attempts to connect to Chrome, even when:

• Microsoft Edge is the system default browser
• The Claude browser extension is installed and active in Edge
• The native messaging host is correctly registered for Edge in the Windows registry (HKCU\SOFTWARE\Microsoft\Edge\NativeMessagingHosts\com.anthropic.claude_browser_extension)
• Chrome is not installed or not in use

Current Behavior

1. User activates Cowork mode in Claude Desktop
2. Claude Desktop attempts "Get tabs" operation
3. Dialog appears: "Check your Claude in Chrome account" with "Open Extension Settings" link
4. Clicking "Open Extension Settings" targets Chrome, not the default browser
5. The feature is unusable for Edge-only users

Expected Behavior

• Claude Desktop should detect which Chromium-based browsers have the extension installed
• At minimum, support Chrome and Edge (Edge is the default browser on Windows with a large user base)
• The dialog text should be browser-agnostic or reference the correct browser
• "Open Extension Settings" should open the correct browser's extension page

Environment

• OS: Windows 11 Enterprise 10.0.26100
• Claude Desktop: Latest (via Microsoft Store - Claude_pzs8sxrjxfjjc)
• Default Browser: Microsoft Edge (confirmed via registry MSEdgeHTM)
• Edge Extension: Claude extension installed (fcoeoabgfenejglbffodgkkbkcdhcgfn)
• Registry: Native messaging host IS registered for Edge

Registry Proof

HKCU\SOFTWARE\Microsoft\Edge\NativeMessagingHosts\com.anthropic.claude_browser_extension
    (Default) REG_SZ C:\Users\...\Claude\ChromeNativeHost\com.anthropic.claude_browser_extension.json

Screenshot

The Cowork dialog hardcodes "Chrome" in the message text:

⚠️ Check your Claude in Chrome account The Chrome extension needs to be signed in with the same account as Claude Desktop.

Related Issues

• #24367 (closed as duplicate - Edge native messaging host registration)
• #14391 (closed/stale - Chromium/Brave browser support on Linux)
• #34830 (open - Dia browser support)

These previous issues focused on native messaging host registration. This issue is specifically about Claude Desktop Cowork mode not supporting Edge even when all infrastructure is correctly configured.