When radiation from a colder source arrives at a warmer surface there is some debate about what happens next. To make the question more concrete lets say that the colder source is at temperature 288K. The warmer surface is at 888K and has emissivity of 1.
3 possibilities

1. We ignore such radiation because it cannot happen.
2. The radiation is subtracted from the much larger radiation of every wavelength leaving the hotter surface.
3. The radiation is fully absorbed and its effect is to be re radiated at characteristic temperature of 888K (plus infinitesimally small T increase due to radiation absorption).

I would have thought that 2. and 3 are more plausible than 1.

Both 2 and 3 satisfy the Stephan Boltzmann equation.
3 however seems to imply that the radiation from colder object is transformed into much higher quality radiation and a possible second law of thermodynamics infringement.