It seems https://github.com/pdsinterop/test-suites/blob/master/servers/php-solid-server/cookie/app/index.js#L8 works
But https://github.com/solid/solid-auth-fetcher/blob/master/src/obtainAuthHeaders.ts#L81 doesn't.

The only difference I see is Content-Type instead of content-type as the header name, and that it uses node-fetch directly instead of through CustomAuthFetcher, but at least it works now :)