Are X-ray Clusters Cooled by Heat Conduction to the Surrounding Intergalactic Medium?

We show that X-ray clusters would have cooled substantially over a Hubble time by transport of heat from their hot interior to the their envelope, if the heat conductivity had not been heavily suppressed relative to the Spitzer value due to magnetic fields. The suppression is required in order for the observed abundance of hot X-ray clusters to be consistent with predictions from popular cosmological models. If a similar or stronger suppression factor applies to cluster cores, then thermal conduction can not be the mechanism that prevents cooling flows there.