Nginx reverse proxy to wordpress as a subdirectory

website.conf

server {
    server_name server.com;

    root /var/www/public;

    location /blog/ {
      rewrite ^([^\?#]*/)([^\?#\./]+)([\?#].*)?$ $1$2/$3 permanent;
      proxy_pass http://[IP or website]/;
      proxy_read_timeout 90;
      proxy_connect_timeout 90;
      proxy_redirect off;
      proxy_set_header X-Real-IP $remote_addr;
      proxy_set_header X-Forwarded-For $proxy_add_x_forwarded_for;
      proxy_set_header X-Forwarded-Proto $scheme;
      proxy_set_header Host $host;
      proxy_set_header X-NginX-Proxy true;
      proxy_set_header Connection "";
    }

    location / {
        # try to serve file directly, fallback to index.php
        try_files $uri /index.php$is_args$args;
    }

    location ~ ^/index\.php(/|$) {
        fastcgi_pass unix:/var/run/php/php7.1-fpm.sock;
        fastcgi_split_path_info ^(.+\.php)(/.*)$;
        include fastcgi_params;
        fastcgi_param SCRIPT_FILENAME $realpath_root$fastcgi_script_name;
        fastcgi_param DOCUMENT_ROOT $realpath_root;
        internal;
    }

    error_log /var/log/nginx/project_error.log;
    access_log /var/log/nginx/project_access.log;
}

wordpress.conf

server {
  listen 80 default_server;
  listen [::]:80 default_server;

  root /var/www/wordpress;

  server_name _;

  location = /favicon.ico {
    log_not_found off;
    access_log off;
  }

  location = /robots.txt {
    allow all;
    log_not_found off;
    access_log off;
  }

  location / {
    index index.php;
    try_files $uri $uri/ /index.php?$query_string;
  }

  location ~ \.php$ {
    include fastcgi_params;
    fastcgi_intercept_errors on;
    fastcgi_index index.php;
    fastcgi_pass unix:/run/php/php7.3-fpm.sock;
    fastcgi_param  SCRIPT_FILENAME $document_root$fastcgi_script_name;
  }

  location ~* \.(js|css|png|jpg|jpeg|gif|ico)$ {
    expires max;
    log_not_found off;
  }
}