Theoretically, it should be possible. I have not tested this out myself and so take this with a grain of salt :) You might have to remove the "Is Initial View Controller" setting from the five storyboards so that none of the initial view controllers load - I am not quite sure if this would be needed or not, but might be worth trying.
Then, in your menu, you would instantiate the first view controller from each of your storyboards when the relevant button is clicked and push the new view controller on to your navigation stack, or, simply display the view controller modally, depending on what you want.
I believe the above should work but, as I mentioned before, I have not tested this myself. If you do run into issues, do ping me here and I'd be happy to help.