You don't need the file when using NGINX.
I'm experiencing the same problem when on my test install. On my production server it works as expected and I'm pretty much using the same config except that the test install uses a auth_basic on the whole site... (this could be the issue!)
The following is the path that IPS is trying to access according to the NGINX logs. The post here says that it is invalid https://community.invisionpower.com/4guides/developing-plugins-and-applications/rest-api/sending-your-first-request-r167/
/api/index.php?/core/hello
The config I use is the one that @prupdated mentioned above, IE
location /api/ {
rewrite ^/api/(.*)$ /api/index.php;
}