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This is implied by Einstein in this paper on page 368

http://myweb.rz.uni-augsburg.de/~eckern/adp/history/einstein-papers/1912_38_355-369.pdf where he says:

"Die Ganggeschwindigkeit ω der Gravitationsuhr ist also c proportional, wie dies für Uhren jeder Art der Fall sein soll." (The speed ω of the gravitational clock is thus proportional to c as it should be for clocks of all kinds.)

If this is true, how exactly would we detect a change in the speed of light (if it were variable for some reason)?