I don't think the time rate is changed by gravitation.
There are many kinds of clocks, atomic, gravitational, mechanical, chemical, or even biological. Their time rate varies to bi-direction with gravitation change.
The test should proved light velocity change.
That is, from up to bottom...
GR predict that a photon climbing in the Earth's gravitational field will lose energy and will consequently be redshifted.
A test was done by Pound and Snider in 1965 using the Mossbauer effect
They measured the redshift experienced by a 14.4 Kev rays from the decay of Fe in climbing up...