Yes, he's right. That's one reason why GPS satellite clocks are purposefully set to keep the
wrong time on Earth so they will keep the
right time in orbit. See also
Project GREAT: General Relativity Einstein/Essen Anniversary Test: Clocks, Kids, and General Relativity on Mt Rainier.
Atomic clocks are internally and very precisely temperature controlled. If they were not they would indeed run differently than "proper time" (the time in your current reference frame)
Edited by JonF, : No reason given.