Hierarchical Custom Taxonomies and Conditionals

So.. as always, the trick is to ask “Why does this work how it does?” and look up the root. In this case through cross-reference searching where in_category() is defined, copying that function to functions.php and replacing the standard $category with your custom $page-category.

You’ll find in_category in wp-includes/category-template.php, l. 237.

/**
 * Check if the current post in within any of the given categories.
 *
 * The given categories are checked against the post's categories' term_ids, names and slugs.
 * Categories given as integers will only be checked against the post's categories' term_ids.
 *
 * Prior to v2.5 of WordPress, category names were not supported.
 * Prior to v2.7, category slugs were not supported.
 * Prior to v2.7, only one category could be compared: in_category( $single_category ).
 * Prior to v2.7, this function could only be used in the WordPress Loop.
 * As of 2.7, the function can be used anywhere if it is provided a post ID or post object.
 *
 * @since 1.2.0
 *
 * @uses is_object_in_term()
 *
 * @param int|string|array $category. Category ID, name or slug, or array of said.
 * @param int|post object Optional.  Post to check instead of the current post. @since 2.7.0
 * @return bool True if the current post is in any of the given categories.
 */
function in_category( $category, $_post = null ) {
	if ( empty( $category ) )
		return false;

	if ( $_post ) {
		$_post = get_post( $_post );
	} else {
		$_post =& $GLOBALS['post'];
	}

	if ( !$_post )
		return false;

	$r = is_object_in_term( $_post->ID, 'category', $category );
	if ( is_wp_error( $r ) )
		return false;
	return $r;
}