Shortcut to apply header style 4 (and 5, 6...)
Found it, although I did not expect the solution to be this simple. To do this, follow these steps:


Right-click the Heading 4 style in the ribbon and choose 'Modify' (Or through Layout > Styles > Heading 4 > Modify
In the left hand bottom corner, select 'Shortcut'
Assign shortcut ⌘⌥4
Repeat for Heading 5, 6...
Done!