Render a window shadow on the floating chat input window - #331225

PR classified as team-member.

⚠️ Head commit is NOT signed by a trusted team member, so the full 2-approval requirement applies. (Head commit 81415fc is signed by GitHub, not by a trusted team member. GitHub adds its own "Verified" signature to commits it creates on your behalf — e.g. edits made in the github.com web editor, suggestions applied from a review, or commits made via Codespaces or the GitHub API. That signature only proves GitHub made the commit, not that a trusted team member authored it, so it does not earn the signing credit. To get the credit, create the commit locally, and sign it with a trusted key.)

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⚠️ Collaborator approvals: 0/2.

Needs 2 more collaborator approvals to merge.

✅ No newly added runnable binary files.

✅ No protected engineering system files changed.