Built-in Windows Server Backup of Server 2012 Hyper-V does NOT backup VMs on CSV

This was not entirely clear to me, to say the least.
It seemed that the native Windows Server Backup feature in Windows Server 2012 had added support to backup Hyper-V VMs running on the local machine. But alas, this is not the case when these VMs are stored on a CSV volume.
See this table on the article about Windows Server Backup in Server 2012 on technet:

Support for backup of Cluster Shared Volumes (CSVs) :
Limited support for backing up CSVs as follows:

1. - Virtual machines hosted on CSV’s cannot be added as part of backup configuration
2. - Windows Server Backup has to be configured on all nodes to ensure that backup and recovery will be available in the event of a failure on one of
   the nodes in the cluster.
3.  - Volumes recovery not supported
4. - Security access control lists are not applicable on CSV file service root. Therefore, file recovery to the root of CSV volume is not supported

And indeed, when trying to backup a VM running on a CSV on my Windows Server 2012 machine, i got these errors:

Application backup
Writer Id: {66841CD4-6DED-4F4B-8F17-FD23F8DDC3DE}
Component: 67FF0C4A-E91B-442D-840A-65257B465E45
Caption     : Backup Using Child Partition Snapshot\vm-dc01
Logical Path:
Error           : 80780175
Error Message   : Component was skipped from volume shadow copy.

Detailed Error  : 80780189
Detailed Error Message : Windows Backup cannot backup an application with files on a CSV volume.

So, no dice

So, no dice…