Prefix of the parent name is what you should be looking for. Next thing you do is you go to System administration -> Users, look for yourself, click on the record with right button and press Record Info. Here you press Rename and type UserId that you need.
After you've done that, you should reopen AX client, and voila, you can access those private projects. After doing operations with those projects, you can change your id back.The only issue, when there is a user with this UserId already in the system. In this case you could change his userId first, and in the end change it back.
Actually is easier to look into ModelElements directly on sql
SELECT *
FROM [MicrosoftDynamicsAX_model].[dbo].[ModelElement]
where [ModelElement].Name like '%projectName%'
Then is just change the ElementType (38 for private, 37 for shared) to shareproject
update [MicrosoftDynamicsAX_model].[dbo].[ModelElement]
set ElementType = 37
where [ModelElement].Name like '%projectName%'
And thats it
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