The default setup generated the following error on any setup of permalinks other than the first, plain one.

The response is not a valid JSON response.

Unfortunately the plain one has question marks in it. Which means that it won’t get indexed well.

Finally one obscure web result. It said “Fix the nginx wordpress rules”.

location / {
index index.php index.html index.htm;
try_files $uri $uri/ /index.php$is_args$args;
}

And indeed, putting this in the configuration file on /etc/nginx/sites-available/socho fixed it! Yay.