Agreed the stated claims don't seem to make much sense. Using a point mass 1 meter away and (G*M)/(r*c^2) I'm getting that you'd have to stand next to the clock for ~61 years to cause a time dilation due to gravity exceeding 10^-16 seconds.
Actually, it's even worse than that: the design of the clock makes it so that the cesium atoms doing the actual time keeping are in free-fall while they are being observed. So it is physically impossible for any gravitational influence to change the accuracy of the clock.
