Well it’s a mess, isn’t it. Plenty of conflicting information from too many sources. FAA makes reference to applying temperature corrections , as does PANSOPS. Other sources don’t appear to and there seems to be some conflict between the meanings of compensation (FMC) and correction.
Intuition, experience and logic would say that if a known error can be corrected for then it should be, so in the meantime I’ll continue to use my airline’s SOP which is to apply temperature error correction for all approaches when the airfield temp drops below 0.
Stay safe, all