Feat Request: sphinx-book-theme (pydata fork)

You guys support pydata, but sphinx-book-theme is a very popular fork (many enterprise folks use it, including _Unreal Engine_ docs) - would love to see official support.

Our [source-available] docs for Xsolla Backend at https://docs.goxbe.io also use this theme.

Would love to see official support! In Sphinx 8, here's what we currently do to make it work:

1. Fix this breaking bug: https://github.com/algolia/sphinx-docsearch/issues/60

2. Add a `algoliaDocSearchSphinxBookThemeSupport.css` to adjust container z-index and ensure the theme search is hidden:

```css
/*
##################################################################################
- Initial template suggestion from yafimvo @
https://github.com/jupyter-book/jupyter-book/issues/1324#issuecomment-1512757574
##################################################################################
*/

/* -- sphinx-book-theme Support --------------------------------------------------- */
/* Hide the old theme's (!Algolia's) search wrapper window when hitting Ctrl+K */
.search-button__wrapper, #search-input {
display: none !important;
}

/* Make sure Algolia's search container is always on top */
.DocSearch.DocSearch-Container {
z-index: 10000;
}
```

That's pretty much it, iirc!

Review the Sphinx 8 compatibility issue linked in the report and the proposed algoliaDocSearchSphinxBookThemeSupport.css contents. Compare the existing pydata theme support with sphinx-book-theme behavior, then verify that the theme works with the reported CSS adjustments and the breaking bug resolved.

Written by the indexing model from the issue text.