Deep link into the search UI

[dannysheridan]

Feature Description

It would be valuable to support deep linking into the documentation site where:

A search modal automatically appears upon page load.
A pre-populated search query is passed via the URL parameters to display results directly in the search interface (either AI search or docsearch).

Use Case

The AI chat on https://elevenlabs.io/docs is sweet. I want to show a friend AI chat by sharing a link.

Proposed Experience (Optional)

https://docs.example.com/page-path?searchType=ai
https://docs.example.com/page-path?searchType=semantic
https://docs.example.com/page-path?searchType=ai&query=API%20integration
https://docs.example.com/page-path?searchType=semantic&query=API%20integration

Alternatives Considered (Optional)

No response