Server timeouts and other errors when exporting

Every time I export a post of any length I get a server timeout error or an unknown error message. I end up with a draft post with the correct title but no content. I tested with 4 test documents.

1. A trivial doc with only one line and no images. This works fine, and takes about 10 seconds to export.
2. A trivial doc with only one line and one image. Still works, but takes about 30 seconds to export.
3. A real doc, with thousands of words and several images of various sizes. This one fails. All posts of substantial length with substantial number of images that I have tried fail.
4. A copy of the the same doc as (3), with all images removed. This failed the first time I clicked save, with the error "Error Your server returned an unexpected response. This could be a misconfiguration, a bad plugin, or something may be broken. Don't panic yet though!". I clicked save again immediately and it succeeded. Took about 20 seconds to export.

I am happy to provide you with access to all of these test docs, and possibly even to our site backend (temporarily) if it would help debug. But I prefer you contact me directly to discuss that. If that would help, let me know and I'll give you contact details.

Start by reproducing exports using the four document cases described in the issue, especially the long document with several images, and record the server response and elapsed time. Inspect the export path and the WordPress publishing request; done means substantial posts export without timeouts or empty drafts, while preserving the successful small-document cases.

Written by the indexing model from the issue text.