I'm using SharePoint Server 2013 and I need a powershell script that published a custom user profile property to the User Information List.
You should just need to mark the User Profile Property as Replicable. If you can't get it let me know and I'll post code as an answer. You also have to update content on the target site collection? Your users have to be "Active" for the replication to work.They have to interact with the site by adding content or some other interactions. There is a hidden flag on the UIL that is set when a user is "Active", then the sync jobs update the info.