To cut a long story short:
Whatever the thermodynamic effect of a colder object radiating on a warmer object is, it has already been taken into account by the coefficient of thermal conductivity which, despite its name, measures all kinds of diffusive heat transport including radiation. (How could it not?)
That is the correct solution. I could also mention that I am not alone with this opinion.
For example, G&T write the following in their first falsification paper:
"A physicist starts his analysis of the problem by pointing his attention to two fundamental thermodynamic properties, namely
the thermal conductivity, a property that determines how much heat per time unit
and temperature difference flows in a medium;"
In their reply to Halpern et al. they write:
"Speculations that consider the conjectured atmospheric CO2 greenhouse effect as an "obstruction to cooling" disregard the fact that in a volume the radiative contributions are already included in the measurable thermodynamical properties, in particular, transport coefficients."
I couldn't agree more. What I have just stated is very powerful in its simplicity, since, if you want to know the thermodynamic effect of doubling the CO2 concentration you only need to measure the changes in the transport coefficients. These changes will of course be unmeasurable (although there is probably some tiny factual difference). And that's it. No need for any redundant radiative transfer calculations. The Greenhouse Effect is no more, gone like a fart in the wind.
The reason I am mentioning this is that there is a tendency of some people to over-do things. The overall theme of these various claims is that radiation from a colder body cannot be absorbed and/or cannot have any effect on a warmer body. My reaction to that is: Why not? Look at Newton's law of cooling:
The heat tranfer Q from hot (T1) to cold (T2) is given by
Q = k(T1 - T2)
Since the temperature of the colder object T2 occurs in the formula the colder object must be doing something with the warmer object. If it did nothing we wouldn't feel the difference between 20 and -10 degrees. What is this something? Well, maybe in part it is the absorbtion of radiation. I don't know for sure but some people seem to know a whole lot about those things. And if it isn't radiation it must be something else. Does this "something else" violate the second law?
There is no possibility to discuss all of the excessive staments here, I have already done so to a certain extend previously on this blog. I just want to point out an obvious danger.
Accepting that a warmer object can indeed absorb radiation from a colder object and that this might slow down the cooling is not the same thing as accepting the Greenhouse Effect.
If you do claim the contrary then the Lukewarmers have won. Then their "trademark" has been saved for future generations and can pop up any time with some new twisted definition. Insisting on this naïve simplification would be a great disservice to society.