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Answers to What aspects of reentry heating 'scale as the 8th power'? including my unnecessarily downvoted answer there explain that radiative heating from the hot plasma to the spacecraft are an extremely strong function of relative velocity between the spacecraft and the atmosphere, which rotates with the rest of the planet.
A 10% difference in reentry speed could double the radiative heating load on a spacecraft entering Mars' atmosphere ($1.1^8=2.14$).
Mars' standard gravitational parameter $GM$ is 4.2828E+13 m^3/s^2 and a 111 km Mars orbit has a radius of 4000 km. The orbital speed is
$$v = \sqrt{GM/a} = 3272 \ \text{m/s}$$
and the atmosphere's rotational velocity at that altitude is about 241 m/s, so the ratio of radiative heating prograde versus retrograde would be
$$\left( \frac{3272 + 241}{3272 - 241} \right)^8 \approx 3.3$$
and that's a big difference!
Radiative heating is a huge problem and so ablative heat shields continuously vaporize, producing a layer of gas which is somewhat opaque to the thermal infrared light radiated by the plasma so that much of it never reaches the spacecraft, but nothing is 100% effective so you'd still like to keep heating to a minimum by entering the atmosphere at the lowest possible relative velocity.
Note that like the Space Shuttle, the SpaceX Starship planned to land on Mars will use ceramic tiles and not an ablative heat shield, so the outside surfaces of the tiles will be directly exposed to full radiative heating.
The tiles will (hopefully) have high reflectivity for thermal infrared, and low thermal conductivity so that the thermal loading on the ship's structure behind the tiles is low. It will also use aerodynamic fins to somewhat reduce its rate of descent so that it can dump more velocity before reaching denser parts of the atmosphere.
Still, with no ablative heat shield to block radiative heating, it will certainly benefit from the reduced relative velocity of entering with Mars' atmosphere's rotation rather than against it.

Source: Lecture #1: Stagnation Point Heating