We can set the visibility property to be dependent on the user's rights to the record in question by following these steps:

1.  From the tab in question, toggle on design mode
2.  For the KPI in question, click properties
3.  Select the Parameters field under the UI Widget subgroup
4.  Select KPIINSTANCEID and select edit
5.  Copy the GUID from the value field
6.  Click cancel
7.  Clear the visible property
8.  Select expression for the visible property
9.  Set the expression to "=Globals.CurrentAppUserGrantedKpi("[GUID]")"
10.  Click ok
11.  Click save

If the user doesn't have rights to any items on the tab, the tab will no longer be visible.