ChatGPT tells me that the time difference between the top of mount everest and sea level is in the nano second range - so completely dwarfed by network latency and maybe doesn't matter?
If you were to supply that answer in a group concerned with time and gravity they might ask you if that was the right question to ask wrt "greatest relativistic time seperation between two points on 'the earth's surface'"
ChatGPT likely won't help - but you could look into the fact the the earth isn't round, being an oblate spheroid with 20km less radius at the poles than the equator.
Of course the fact the ideal WGS84 ellipsoid, the official mean global sea level, and the geoid (gravitational surface of equipotential) don't all align must surely come into play here - and that bloody great "gravitational hole" somewhere south of Ceylon.
Pure speculation on my behalf though.