Change the DC in an ADLDS application directory partition name (the Naming Context)

Question

I have a server on which I used the Active Directory Lightweight Directory Services Setup Wizard to create a new application directory partition:

I need to rename the DC part of the Partition name from the screenshot above.

The Partition name is the Naming Context for all my ADLDS objects - when I connect to it using ADSI Edit, for example, the connection string is:

LDAP://localhost:389/CN=SomeValue,DC=Example

So every object in that Naming Context has CN=SomeValue,DC=Example as part of its distinguishedName.

Is there a way that I can simply update everything in this application directory partition to be as if I had originally put CN=SomeValue,DC=TheCorrectValue as the Partition name / Naming Context?

Note: The AD LDS instance / directory partition is not complex at all... the only thing I've done since creating it is add user objects via LDAP with C#. The reason I don't just start over with a new application directory partition is that the user objects are currently being used to authenticate users to an application.