Show lowest simple product price in category page

I want to display the lowest simple product price in the content-oroduct_cat.php page. The following code from Fancy Squares works for showing the lowest price but I want to show Simple products only, i.e. omit grouped products.

/* SHOW LOWEST PRICE ON CATEGORY PAGE */
//woocommerce get lowest price in category
function wpq_get_min_price_per_product_cat($term_id)
{    
    global $wpdb;

    $sql = "
    SELECT  MIN( meta_value+0 ) as minprice
    FROM {$wpdb->posts} 
    INNER JOIN {$wpdb->term_relationships} ON ({$wpdb->posts}.ID = {$wpdb->term_relationships}.object_id)
    INNER JOIN {$wpdb->postmeta} ON ({$wpdb->posts}.ID = {$wpdb->postmeta}.post_id) 
    WHERE  
      ( {$wpdb->term_relationships}.term_taxonomy_id IN (%d) ) 
    AND {$wpdb->posts}.post_type = 'product'  
    AND {$wpdb->posts}.post_status = 'publish' 
    AND {$wpdb->postmeta}.meta_key = '_price'
    ";

    return $wpdb->get_var($wpdb->prepare($sql, $term_id));
}

I tried using :

AND {$wpdb->posts}.product_type = 'simple'

but this didn’t work. How would I display only simple products?

Read more here: Show lowest simple product price in category page

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