I don't think this is true.

No sure about General Relativity but, I did some special relativity transformations in my final year of high school.

First off all we need to consider that there is no absolute velocity or reference frame, so we just need pick a stationary point and calculate rotational speed of the earth's core as a velocity relative to an observer on the earths surface.

So if we consider the earths surface as the stationary point, then that means the earths rest age (T₀) age when V=0 to be 4.543 Billion years, then according to the Lorenz transformation the the earths core age relative to us as stationary observers will be (T₀)* γ where  γ=the Lorenz transformation is equal to √(1-(v2/c2)) and v is the orbital velocity of the earths core and c is the speed of light (300000000m/s)

The earths inner core has a radius of 1216km and its rotational speed (v) is just 2πr/t which is 463.31 m/s. So the lorentz transformation for this velocity is virtually 1 (since v2/c2 at that speed is 0.000000000000238 which is virtually nothing) meaning almost no relativistic effects occur at such a low speed and the age of the earths core is pretty much the same as the surface because the there isn't a significant enough speed difference between the 2 to cause time dilation of one relative to the other.