There are two conjectures here. The first is that 73 is the only number that matches these criteria. The second (as stated by Sheldon) is that meeting the criteria make a number "the best number". The second conjecture... let's just call that "unproven".
I would call that second one a definition, not a conjecture. "Best" is an arbitrary value judgment. You can't prove what is or isn't best because it is not an objective question.
My pet peeve with 73 being "best" number is that a lot of these properties relies on representing this number in decimal (or binary in case of Sheldon-Norris number). I'd like properties that were independent of base selected to represent the number.
https://math.dartmouth.edu/~carlp/sheldon022119.pdf
"The episode of The Big Bang Theory that contained the discussion about 73, was actually the 73rd of the series!"
But yes, it is an open question.