I think the supposition is based on the differential equation for pressure of column of gas as:
dP/dr = density*G*M/r^2
Then, further assuming temperature and composition are constant, density is proportional to pressure. Then, for mars, GM/r^2 at surface (surface gravitational acceleration) is lower than for earth; thus at given pressure, dp/dr will be lower than for earth. Thus, if pressure at surface matches earth, pressure and density above surface will higher (slower rate of decline).
Big flaw: even if you assume T and P are the same on the surface of Mars as for earth, there is absolutely no reason to believe that T as a function of altitude will be the same, let alone constant. Further, one expects composition to vary with altitude. However, I wouldn't be surprised of the broad conclusion were true: that matching P and T at the surface will lead to higher density at given altitudes compared to earth.
(To clarify the more complex realistic case, P = density *specific gas constant * T. Realistically, both T and specific gas constant (which is dependent on composition) vary with altitude).