By default the plugin tries to generate the CSS into the wp-content/plugins/ajax-search-pro/css/style.instances.css file directory. If this directory has wrong permissions or not writeable, then as a fallback it has to load it into the header as inline CSS – it does NOT affect SEO as the inline CSS is valid as long as it's presented in the head section of the HTML code.
To remove the inline CSS from the head, make sure that the wp-content/plugins/ajax-search-pro/css/ directory is writeable – by chmodding it to 755 or 775. If that does not help, then the www-data process might not have the ownership of the directory, and you will have to adjust it via SSH.
Also, make sure that the Force Inline Styles option is turned off on the Compatibility settings submenu: https://i.imgur.com/C1eQXuc.png
IMPORTANT: If you sucessfully changed the permissions, then save the Compatibility settings submenu, to have the stylesheets re-generated.Best,
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