Multisite: Only Drupal Root recognized for subpath
Adding a rewrite rule into the .htaccess file fixes the problem. Without the rewrite rule for this path, the default rewrite rule goes into effect, pointing to the example.com/index.php instead of example.com/sub/path/index.php. 

Disadvantage: I will have to patch .htaccess every time I upgrade Drupal core. 

RewriteCond %{REQUEST_URI} ^/sub/path/(.*)$ 
RewriteRule ^(.*)$ /sub/path/index.php [L,QSA]