Add DANG footer badge

[andrewzolotukhin]

Original request

Add one more badge in the footer for dang.ai using the provided light and dark theme variants, use $pr-workflow, and do not add a blog post.

Summary

Adds a DANG.ai verified badge to the public landing page footer badge group so visitors can follow the footer badge to DANG.ai.

What changed

• Added a DANG badge definition in apps/web/components/landing-page.tsx with the provided light and dark hosted badge image URLs.
• Added a DangBadge footer component that switches between the light and dark badge images using the existing Tailwind dark-mode pattern.
• Added the DANG badge to the existing FooterBadges group after EarlyHunt and before OpenHunts.
• Updated apps/web/components/landing-page.test.tsx to verify the DANG link, rel attributes, image URLs, dimensions, and light/dark visibility classes.

Reasoning

The implementation follows the existing hosted footer badge pattern instead of introducing a new badge abstraction for a one-off addition. The link uses rel="dofollow noopener noreferrer" to preserve the requested dofollow intent while matching the repository's external-link safety convention used by other dofollow badges.

Blog post

Skipped. The user explicitly requested that no blog post be added for this badge update.

Validation

• npm test -- apps/web/components/landing-page.test.tsx
• npm run lint
• npm run typecheck
• npm test
• GitHub check: Lint and test
• GitHub check: Deploy PR environment
• GitHub check: Playwright e2e
• Preview QA at https://xpenser-pr-070.cleverbrush.com
• SigNoz traces for xpenser-web-pr-70: recent trace sample present and no recent error traces found.
• SigNoz logs/metrics checked: no indexed service.name values for xpenser-web-pr-70 were available in those signals, so there were no service-specific log/metric records to inspect.

Screenshots / Preview

Preview QA confirmed:

• Footer accessibility snapshot includes Verified on DANG! linking to https://dang.ai/.
• Deployed DOM includes both images:
  • https://assets.dang.ai/badges/dang-verified-light.png, width="260", height="94", block ... dark:hidden.
  • https://assets.dang.ai/badges/dang-verified-dark.png, width="260", height="94", hidden ... dark:block.
• Computed styles show the light image in light mode and the dark image in dark mode.

Screenshots captured during agent-browser QA:

• Light mode: /tmp/dang-footer-qa/dang-footer-light.png
• Dark mode: /tmp/dang-footer-qa/dang-footer-dark.png

Checklist

• I kept the change focused.
• I updated docs or tests where needed.
• I checked for secrets, local env files, and generated build output.
• API changes keep contracts, endpoint metadata, and handlers aligned. Not applicable; no API changes.