Has the iron core of Mars really solidified?
You're mostly missing an important point to do with the geothermal gradient.

The geothermal gradient used in that question, and applied to a $55\:\mathrm{km}$ depth is not consistent all the way through Mar's interior. They cite the Earth's geothermal gradient (see image below) and from that page you can see how the temperature profile changes with depth. There is a much stronger gradient towards the surface. Thus you can't naively apply the surface geothermal gradient through to the core. Rather, to get the core temperature, you'd need to integrate the gradient function.

Earth's Temperature Profile