Ordering Current Post Taxonomy Terms

Hello,

I would like to organize the current posts taxonomies in a list with parent – child taxonomy terms.

For instance, I would like a structure like this:

Term
-Term Child
–Term Grandchild
—Term Great Grandchild
Term
-Term Child
–Term Grandchild
—Term Great Grandchild

Here is the code I have been working with, but have come to a roadblock trying to achieve this. Thanks in advance.

<ul style="
    font-size: 20px;
    -webkit-column-count: 4;
    -moz-column-count: 4;
    column-count: 4;
	text-align: left;
"><p style="font-weight: none; font-size: 10px;"><?php
$args = array('orderby' => 'term_group', 'order' => 'ASC', 'fields' => 'all');
$product_terms = wp_get_object_terms( $post->ID, 'zip-code' );
if ( ! empty( $product_terms ) ) {
	if ( ! is_wp_error( $product_terms ) ) {

			foreach( $product_terms as $term ) {
				echo '<a href="' . get_term_link( $term->slug, 'zip-code' ) . '">' . esc_html( $term->name ) . '</a> , ';
			}

	}
}
?></p></ul>