Defining high z-index value for the elements that were covered by the tabs i.e. for the list items didn't help. While searching for correct solution I saw multiple complains about very similar behavior so I decided to post my solution.
If you encounter a similar problem the solution is quite simple - set the high z-index value not for the menu items but for their parent container! In my case it was the entire Custom Menu div. If it doesn't work with the direct parent work you way up in HTML DOM to find the right one.