As far as national championships, there is no doubt of this. However, most of those were awarded by some valid entity at the time, even if their importance has diminished over time.
This seems different to me. I may be wrong but I can not remember any team claiming a conference title when they a) lost the title game for one of the years and b) did not even play in it for the other. If there was no title game to play, and they had the same record as the recognized champ, then maybe I could see an argument, but that is not the case here.
Big difference between claiming a title that was awarded by Freds Sports Authority in 1944 and what they are doing with the conference (division but not specified) titles IMO.