You know this how?
Churchill, for most of the time the history of WWII was being written, was in the opposition, eying a political comeback.
Reviews I've read of Churchill's work - again, I haven't read the volumes - suggest that Churchill pitched the narrative (well, his ghostwriters did) to his own need in the present more than anything - earning money, presenting his leadership during the war as a triumphant success.
These reviews could be wrong, of course, so how do you know that Churchill was concerned, as Fauri says, "that there existed no satisfactory evidence to substantiate public claims that execution gas chambers did indeed exist" rather than omitted and included based on other goals? What's the evidence, aside from Fauri's argument that it must have been so because it was so?
And the second part was actually edited by the opposition. Who were more privy to what would still fall under the official secrets acts, and what could be divulged at time of publication.