Polylang compatibility issue

Hello, I have a problem with Polylang and your extension.

The downloads are not correctly translated.

On Polylang’s recommendation, I even added this snippet:

add_filter( 'pll_get_post_types', 'add_cpt_to_pll', 10, 2 );
 
function add_cpt_to_pll( $post_types, $is_settings ) {
    if ( $is_settings ) {
        unset( $post_types['dlm_download'] );
    } else {
        $post_types['dlm_download'] = 'dlm_download';
    }
    return $post_types;
}

To be sure that the CPT is added correctly.

This does not change much, because even if it is linked to another language on the single download

• The flags do not appear in the download list (wp-admin/edit.php?post_type=dlm_download)
• On the front, all languages ??are displayed for the same document

Could you please see what is happening?

(The fault appears on an old site from WPML, but also on a new installation, with Polylang Pro)