OK, I see what you mean, the kinetic energy on average is always non-zero, but the average velocity is zero. There is no other explanation unless you are willing to assign the velocity and momentum a non-zero average value. You can say "the particles have no average velocity but they travel towards the fringes on average". Since this is your explanation then I would ask you to break the mathematics down to something which evaluates areas that are smaller than the uncertainty so you can see each part of the distribution changing by itself. But this is not possible in QM. Although your explanation in terms of the hamiltonian does suffice as an explanation. Your opinion seems to be that the transverse directional motion of the particles is averaged to zero, but still each particle moves, and that the combined movement creates the fringing that is observed. And then it is natural to assume that the "slowing" of the particles is the energy loss that makes up for the energy gain in average gravitational potential. I would say that this is an eloborate explanation that is worth testing, because most physicists do not have the interpretation that would allow for "movement of the particles" prior to measurement.