Hi,
Microsoft doesnt recommend any third-party provider or third-party solution. Microsoft cannot guarantee or endorse the accuracy of any information or of any solution that is presented by Microsoft or by any mentioned third-party provider.
By the way, the whole idea of the archive mailbox is to keep your "online" mailbox with recent / important mail small and accessible on and offline. The archive mailbox is intend to store older / less important mail which is not necessary to access immediately.
Think what will happen if you have a larger archive in outlook and you will cache it locally your offline file will grow big time and outlook will become slow again. One of the reasons to use an archive mailbox is to prevent the grow of offline files.
Best Regards.