BEARenstain?

Okay, I just learned about this conspiracy recently. I always thought it was "BERenSTEIN" as a kid, then I was proven by the theory that it's actually "BEARenSTAIN" only to find it to be "BERenSTAIN" which is inconsistent with both. Do we just suck at remembering how to spell this name, or has the spelling been changed not once, but twice?