Here is a better IUCN source which clearly details historic range in desert areas. I added this to the original post.
AFAIK, the only plains bison in Eurasia and Alaska tundras are introduced animals, meaning they do not have historic presence there. Now, I said this is highly debatable, because there is new evidence that suggests plains bison are much more closely related to the ancient steppe bison than previously thought, but I don't know enough about that subject, and when I know I'm not properly informed, I try not to comment or take a side one way or another.