We are not using cursive here, so Roman cursive is not all that relevant.

I sort of see what you mean here though:

"And since our lowercase letters are based on cursive forms, it makes much more sense to use u in the lowercase"

But I would conclude that the textbox should only use capital letters. When I wrote that it should say "Lvtetia", I obviously meant "LVTETIA". Minuscules are not different letters, they are variant forms, so by using modern capitalization I didn't mean to imply anything about how the program should do it, bicameral writing is a modern fad.