edit_comment capability for subscriber

I have given my site’s subscriber role the following permissions (unserialized DB-query):

[subscriber] => Array
    (
        [name] => Reader
        [capabilities] => Array
            (
                [delete_comment] => 1
                [edit_comment] => 1
                [level_0] => 1
                [quick_edit_comment] => 1
                [read] => 1
                [reply_comment] => 1
                [unfiltered_html] => 1
            )

    )

I want to make the subscriber be able to edit their comments, but I always get an error “Your are not allowed to edit this comment.” I tried the following in debugging (in my comments.php template file):

$link = get_edit_comment_link($GLOBALS['comment']->comment_ID);
$perm1 = current_user_can('edit_comment', $GLOBALS['comment']->comment_ID);
$perm2 = current_user_can('edit_comment');
var_dump($link);
var_dump($perm1);
var_dump($perm2);
global $current_user;
var_dump($current_user->allcaps);

giving the following output (explanation added beside):

(get_edit_comment_link): NULL
(cur_user w/ post-id): bool(false)
(cur_user w/o post-id): bool(false)

(allcaps): 
array(9) { ["delete_comment"]=> bool(true) ["edit_comment"]=> bool(true) 
["edit_posts"]=> bool(true) ["level_0"]=> bool(true) ["quick_edit_comment"]=> bool(true) 
["read"]=> bool(true) ["reply_comment"]=> bool(true) ["unfiltered_html"]=> bool(true) 
["subscriber"]=> bool(true) } 

As you can see, the allcaps shows that the user is allowed to edit_comments. But get_edit_comment_link stays empty, although the role has the capability AND the current_user is the author of the comment.