How to set default status to custom post type

When I save my custom post_type ‘product’, I’d like to set their status to my custom status ‘incomplete’.

Hook wp_insert_post_data filter to force a post status to be set as incomplete before it can be set as published. With the following code only a post that is set as incomplete can be saved:

add_filter( 'wp_insert_post_data', 'prevent_post_change', 20, 2 );

function prevent_post_change( $data, $postarr ) {
    if ( ! isset($postarr['ID']) || ! $postarr['ID'] ) return $data;
    if ( $postarr['post_type'] !== 'product' ) return $data; // only for products
    $old = get_post($postarr['ID']); // the post before update
    if (
        $old->post_status !== 'incomplete' &&
        $old->post_status !== 'trash' && // without this post restoring from trash fail
        $data['post_status'] === 'publish' 
    ) {
        // set post to incomplete before being published
        $data['post_status'] = 'incomplete';
    }
    return $data;
}