The category parameters does not work with custom taxonomies. You need to use a tax_query
instead.
In your current code, replace (which is in any case also wrongly used, category_name
takes the slug, not name)
'category_name' => $category->name
with
'post_type' => 'case-studies',
'tax_query' => array(
array(
'taxonomy' => $tax
'terms' => $category->term_id,
'include_children' => false
)
),