This paper: http://tycho.usno.navy.mil/ptti/1996/Vol%2028_16.pdf
states that "the time rate is appropriate to observers on the surface of the rotating earth, that is, in the ECEF". I'm interpreting that to mean that the issue raised in the OP is not correct, but I am by no means an expert in relativity.
Interestingly, the paper also states ('Missing Relativity Terms?', pp. 195-197) that there has been confusion in the past caused by people thinking the time is measured in the ECI frame. It shows that the uncorrected-for relativistic effects have an error on the order of only 2-3mm for a stationary observer on the earth's surface (the same is not true for e.g. other satellites). 'In short, there are no "missing relativity terms."'
Interestingly, the paper also states ('Missing Relativity Terms?', pp. 195-197) that there has been confusion in the past caused by people thinking the time is measured in the ECI frame. It shows that the uncorrected-for relativistic effects have an error on the order of only 2-3mm for a stationary observer on the earth's surface (the same is not true for e.g. other satellites). 'In short, there are no "missing relativity terms."'