[BUG] Claude Desktop "Public Share" URLs return 403 when fetched by Claude Code

Description

Public share links generated from the Claude Desktop app return HTTP 403 (Forbidden) when Claude Code tries to fetch them via WebFetch. This breaks a natural workflow: sharing a conversation from the Desktop app and then having Claude Code read/reference it.

Reproduction Steps

1. Open a conversation in the Claude Desktop app
2. Use the "Share" feature to generate a public link (e.g., https://claude.ai/share/<id>)
3. In Claude Code, ask it to fetch the shared URL
4. Claude Code's WebFetch tool returns 403 Forbidden

Example

$ curl -s -o /dev/null -w "%{http_code}" https://claude.ai/share/4b3a1a9d-4f5a-4ca6-ba4f-76ffc3f29971
403

Expected Behavior

Claude Code should be able to read public share links so users can reference Desktop app conversations as context — e.g., "continue this work" or "review what I discussed here."

Actual Behavior

The server blocks non-browser requests with a 403, likely via user-agent filtering or a JS challenge. The link works in a browser but is inaccessible to Claude Code.

Why This Matters

Sharing context between the Desktop app and Claude Code is a frequently requested workflow (see #13843, #18803, #28307). Public share links could already serve as a lightweight bridge, but the 403 block prevents it.

Environment

• Claude Desktop (macOS)
• Claude Code (CLI)