I think @timmdeeg is referring to my point, expanded as follows: Choose any interim event whatever. There is a constant axial coordinates geodesic through it, that maximizes proper time from this event to the singularity. There is also a null geodesic in the + axial direction from this event to the singularity (this is a 45 degree line to the right in a kruskal diagram). Proper time along this is obviously zero. Paths in between these lines will have intermediate proper time to the singularity, thus less than maximal. You can understand this as being too close to a light like path.