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@glemaitre @adrinjalali I think this is ready for review. I briefly went through the website and did not see fundamental problems. After the new web theme is deployed, I may:

• Revisit the example gallery pages. Some problems are: empty secondary sidebar (should be fixed when pydata-sphinx-theme==0.15.3 is available); moving the links takes a certain amount of time and may do duplicated work in case some pages are unchanged; only works best for fresh full build because it currently uses a very dumb way of searching for files to be tweaked. I think I will go for FEA Integrate download links and binder/juputerlite buttons with pydata-sphinx-theme secondary sidebar sphinx-gallery/sphinx-gallery#1258.
• There are many unexpected blockquotes to be cleaned up, for instance, bisecting_strategy in this API page. I may later fix gradually.

Please feel free to navigate through the built artifact and let me know if there are problems that I overlooked.

I'm fine that we have a couple of glitches that will be removed soonish.
Fixing upstream is more important because it will impact more people. That's nice.

FYI, we should not block the merge of this PR regarding this #29037 (comment)

This can be tackled in a subsequent PR and I think that this is more valuable to have the website ready for 1.5

Sure (I'm sorry I wasn't at home so did not provide a timely update). Then shall we merge this one rn then I can quickly update #29038 to get the website ready?

Don't worry, I'll ping a second reviewer but I'm sure that we will be able to do it for 1.5 :)
ping @jeremiedbb or @ogrisel

I just realized that I forgot "not" in my previous sentence.

There might be something wrong here: has this PR deleted doc/modules/classes.rst by accident or on purpose?

It was done on purpose. It's now listed in doc/api_reference.py

There might be something wrong here: has this PR deleted doc/modules/classes.rst by accident or on purpose?

Nop, now this file is generated with the doc/api_reference.py file. We will add an entry in the documentation to explain it.

Oh, I see then. Sorry, it seems I was looking for it to show up in some diff in the wrong place.