Radiative Energy Transfer with Filters and Stokes/Anti-Stokes Coatings

An ideal blackbody resides inside a cavity in a medium (heat bath) of temperature T that is also an ideal blackbody. If the small black body is initially at temperature T , could its temperature evolve to an equilibrium temperature T ′ different from T if the inner surface of the cavity were covered with a passive filter? For example, the filter might be an ideal transmitter for light of some wavelengths, while being an ideal blackbody for all other wavelengths (meaning that the filter is perfectly absorbing at the other wavelengths, and emits light with a blackbody spectrum, described by a single “temperature”, at those other wavelengths).