An app host's /__forge/ door lists its open pulls with their check word, linking each to its preview #6073

Last web part of closed pull #5988.

Done when:

• [ ] For a viewer who can read the repository, /__forge/ (src/apps/door.ts) shows up to five open pulls as #N title · check <word>. Each links to its preview when that preview is running or asleep, and to its pull page on the forge otherwise. Every title is escaped.
• [ ] A stranger at a private app still gets the same 404 as for an unknown app.
• [ ] The door's Server-Timing names a pulls phase.
• [ ] A request-level test through the door handler covers a live preview link and a forge link, and bun scripts/design-lint.ts passes.