Using the Moisture Boundary Layer (MBL) model with a fixed lower temperature limit of -1.9C and the approximate average Sea Surface Temperature (SST) of 21.1 C there are a few things that can be inferred but not verified.
With the average radiant energy of 306.9Wm-2 at the top of the MBL and an average measure energy out for the Top of the Atmosphere (TOA) of 239Wm-2, the average radiant energy of all surfaces above or outside of the MBL envelope would be 306.9-239=67.9Wm-2.
By assuming maximum entropy equals 50%, a radiant boundary layer can be approximated at 306.9*0.50=153.4Wm-2. Using the same maximum entropy assumption, a second boundary at 153.4*.5=76.6Wm-2. Ideally, this second boundary would be equal to the first estimate of 67.9Wm-2, there is uncertainty in all of the estimates though. One of the larger uncertainties is the absorption of energy, solar and outgoing long wave, in the atmosphere, another is the energy that gets a free pass through the atmospheric window. So maybe there is another way of looking at the problem?
A second way would be that while the surface is radiating 425Wm-2 inside the MBL, only 118.1Wm-2 is lost at the -1.9C boundary (306.9Wm-2), leaving 188.8Wm-2 (425-306.9Wm-2) unaccounted for in the 50% maximum entropy estimate. 50% of the 188.8Wm-2 would be 94.4Wm-2. This would be the total energy absorbed by the atmosphere from the surface and the sun. That results in to competing choices for the more appropriate radiant boundary layer, 188.8 for the MBLtotal and 153.4 for the MBL outer envelope. Without trying to over complicate this analysis, 188.8-153.4=35.4Wm-2 should be energy that is not doing anything or the "effective" atmospheric window radiation from the MBL. Proving that may be a challenge :)
With an ever so rough estimate of the atmospheric window radiation, 35.4Wm-2, we may be able to work back to isolate the where and what's. By reducing both the MBL surface and outer boundary by 35.4, we get an "effective" surface of 425-35.4=389.6Wm-2 and 306.9-35.4=271.5Wm-2, the difference would be 389.6-271.5=118.1Wm-2 so the MBL energy is still intact, 271.5-35.4=236.1 which is in the ballpark of the TOA emissivity. This implies that MAXENT needs a 35.4Wm-2 fudge factor. Not a permanent fudge, just one to carry until we know where to stick it :)
With the TOA flux at 236.1Wm-2 and assuming the 35.4 is free pass we can use maximum entropy and work backwards again. 236.1-25.4=200.7, 2*200.7=401.4Wm-2 which would be the perfect maximum entropy surface radiant energy if there were no tropopause or energy absorbed in the atmosphere above the surface of course neglecting our PITA free pass energy. Which should be the maximum average surface energy of the planet unless the free pass window closes somewhat. With Ein=Eout at the TOA required, we have a near ideal 200.7 as a greenhouse effect, 306.9-236.1=70.8 as the GHE using the MBL as a surface and the 35.4 PITA free pass. That should make the radiant interaction or OLR absorbed by the atmosphere approximately 70.8-35.4=35.4Wm-2 which implies there may be no free pass radiant window from the true surface but from an average radiant layer inside the MBL, like cloud tops.
I am posting this just in case anyone is following. The object is to have a moist air boundary model and a maximum entropy model which together should highlight any screw ups in each other and provide some sense of confidence as long as they agree, only not perfectly. Where they disagree should be on conductive impacts or sensible if you prefer.