Hotmail/Outlook.com DKIM failure when signing with different domain - header.d ignored

We (sender.org) provide a mail server for a client (example.org) and sign outgoing messages with our private DKIM key (we cannot change the signing domain to the client's domain).

Return-Path: [email protected]
From: [email protected]

The DKIM signature looks like this:

DKIM-Signature: v=1; a=rsa-sha1; q=dns/txt; c=relaxed/relaxed; t=1413987605;
    s=default; d=sender.org;
    h=Reply-To:List-Unsubscribe:List-Id:From:To:Message-Id:Subject:Date:MIME-Version;
    bh=ISuMd/He7ct2h8gGuqNPS6u0Knk=;
    b=nLx/atDvwyl28uB6MSXRUoQO2tH0Dr46wn+IPnxioKMGBHlKFAeEArz0VZyvXIIG
    wM35CG8QspFTsRxvbV3Wfqx1+cR+6RIK1ecILXxCegNd3SCcaMao3fJ5IYAbL4yLiHy
    lbDvXPCSLmJ2uYsNG2ZeIkWDLLOG+WUjyzdtEPD8=

Gmail, Yahoo and others validate the DKIM signature correctly. However in Microsoft mail services such as Outlook.com or hotmail.com the DKIM validation fails:

Authentication-Results: hotmail.com; spf=pass (sender IP is 123.456.78.90) 
[email protected]; dkim=none header.d=example.org; 
x-hmca=none [email protected]

The message says dkim=none even though the DKIM signature is in the header of the email. More importantly, the header.d is wrongly set to example.org whereas it should be sender.org - the domain that is signing.

We have tried to add [email protected] to the DKIM signature and changed the order of the header fields - nothing helped.

Are we missing something?
Is this a bug in hotmail/Outlook.com?
Is there any solution or other indications on how to solve this?

It is not a bug. Its a requirement from Microsoft.

Your emails must have a Author Domain Signature. In other words, domain name of the DKIM signing entity(d=), is the same as the domain name in the author address.

Your author address is [email protected] and signing entity is sender.org