Nginx+PHP-FPM: Let php handle file extensions

I am trying to serve a PHP script that handles some RESTful URIs and to know which format end user needs data, I handled that as the extension in the URI, eg.:

example.com/foo/bar.json?q=x&a=y --> data in ajax format
example.org/foo/bar.xml?q=x&a=y  --> data in xml format

I use Apache httpd + modphp in my development machine and it works just fine, but the stage server uses CentOS + Nginx + PHP. There, the nginx intercepts and tries handling an static json file and returns 404.

How can I prevent Nginx from handling certain file types (eg. json, xml) and let PHP handle those?

My Nginx config:

server {

  # listen [::]:443 ssl http2 accept_filter=dataready;  # for FreeBSD
  # listen 443 ssl http2 accept_filter=dataready;  # for FreeBSD
  # listen [::]:443 ssl http2 deferred;  # for Linux
  # listen 443 ssl http2 deferred;  # for Linux
  listen [::]:443 ssl http2;
  listen 443 ssl http2;

  # The host name to respond to
  server_name example.com;

  include h5bp/directive-only/ssl.conf;
  include ssl/conf/example.com;

  # Path for static files
  root /var/www/example.com/app/public;
  index index.php index.html index.htm;

  #Specify a charset
  charset utf-8;

  # Custom 404 page
  error_page 404 /404.html;

  # Include the basic h5bp config set
  include h5bp/basic.conf;

  # log settings
  access_log off;
  error_log  /var/log/www/example.com/nginx/error/error.log error;

  # turn off access logs and prevents logging
  # an error if robots.txt and favicon.ico are not found
  location = /favicon.ico { access_log off; log_not_found off; }
  location = /robots.txt  { access_log off; log_not_found off; }


  # check if a file or directory index file exists,
  # else pass the request to the index.php as a query parameter.
  location / {
    try_files $uri $uri/ /index.php?$query_string;
  }

  # handle execution of PHP files
  # set php5-fpm socket
  # tell NGINX to proxy requests to PHP FPM via the FCGI protocol
  location ~ \.php$ {
    fastcgi_split_path_info ^(.+\.php)(/.+)$;
    fastcgi_pass localhost:9003;
    fastcgi_index index.php;
    include fastcgi_params;
    fastcgi_param SCRIPT_FILENAME $document_root$fastcgi_script_name;
    fastcgi_intercept_errors off;
    fastcgi_buffer_size 16k;
    fastcgi_buffers 4 16k;
    fastcgi_connect_timeout 300;
    fastcgi_send_timeout 300;
    fastcgi_read_timeout 300;
  }

  # block access to .htaccess files
  location ~ /\.ht {
    deny all;
  }

}

Update: It did work finally, I put the json-related location inside the main / location and changed the $script_file_name for the json-related location to a static script name. Thanks to Tim.