I have several times seen explanations of gravitational free fall (eg, of a small object toward earth with no air resistance) that begin with the following claim about the particles of the free-falling object: its particles that are closer to earth are experiencing greater gravitational time dilation than are the particles that are further from earth. Example of this claim used in explaining gravity: https://youtu.be/UKxQTvqcpSg
I understand that if a clock is at rest at a fixed distance from earth's center, then the clock exphibits greater time dilation the closer it is to earth's center. But the latter scenario is not free fall. And to me, the aforementioned claim seems contrary to the equivalence principle.
I'll put my question this way: Suppose you have a free-falling lab within a small region of spacetime, at an altitude of several km, with no air resistance, and there are two free-falling clocks in the lab, one of them a few nm closer to earth than the other. Will the freefalling clocks undergo gravitational time dilation relative to one another such that an observer in the lab will observe the clocks to be ticking at different rates?
I have no formal education in physics, and I would prefer an answer that is more conceptual and less mathematical, because I may not be able to follow the math very far.