We'd expect:

• Over time, the convection zone should spread its super-adiabatic gradient inwards.

• This would cause convection to diffuse inwards into the radiative zone, resulting in even mixing of the whole Sun.

• Differential rotation should also spread inwards.

...But from the observational evidence:

• We don't see this in our helioseismological results: the core rotates rigidly, without convection.

• We don't see this manifest in the abundance of lithium on the surface.