Why Doesn't Thermal Radiation Render Thermionic Conversion Unable To Approach Carnot Efficiency?

Thermionic Conversion follows the classic Richardson-Dushmann Equation for thermionic current as a function of temperature squared:

$$J_{RD} = A_0 T^2 expleft(-frac{phi}{k_B T}right)$$
where

• $J_{RD}$ is the emission current density.
• $T$ is the emission temperature.
• $A_0$ is a material specific correction factor
• $phi$ is the material’s environment-mediated work function, and
• $k_B$ is is the Boltzmann constant

But thermal power loss is a function of the fourth power of temperature following the Stefan-Boltzmann law:
begin{equation}
P=Asigma T^4
end{equation}
where

• $P$ is the power lost to thermal radiation
• $T$ is the emission temperature.
• $A$ is the area of thermal emission, and
• $sigma$ is the Stefan-Bolzmann constant

Although I see a lot of talk about space charge buildup being the limiting factor on thermionic conversion efficiency, the power of 2 disadvantage from thermal radiation losses seems to make thermionic conversion a fundamentally impractical approach to approximating Carnot efficiency.

What am I missing?