[Plugin: Private WP] Edit so that front page is displayed to non-loggedin users?

I am using the plugin Private WP, whose sole purpose is to keep non-logged-in users out of your blog. I want to modify it to show the home page, but allow access to no other part of the site if you’re not logged in.

Here is the entire code of the plugin:

<?php
function private_wp() {
  if (!is_user_logged_in()) {
    auth_redirect();
  }
}

add_action('get_header', 'private_wp');
?>

I tried changing if (!is_user_logged_in()) to if (!is_user_logged_in() && !is_front_page()), but it didn’t work. Does anyone have any idea as to the nature of the problem?