[math-fun] Two twisted cubes

Consider two 0-centered N-dimensional hypercubes, rotated with respect to each other. Let f(N) = max(over all rotations) min(over all red-blue vertex pairs AB) angle(A0B). For example, f(1)=0, f(2)=45 degrees. This function f is an interesting geometrical quantity. How does this sequence continue? What are its asymptotics? Does f(N) --> 0 when N-->infinity? Or to some positive constant (what)? I know f(4)>=45 degrees. I can prove f(N) > C for some positive constant C (which I could compute) for all large-enough N. I have no upper bound of the form f(N) < C < 90 degrees, but surely one must be true.