I hope this rather soft question is suitable for MO, otherwise please migrate it to MSE.

In their paper defining periods, Kontsevich and Zagier without further comment state that $e$ is (conjecturaly) not a period while many other numbers showing up naturally (conjecturally) are. The former claim is repeated at many other internet sources including wikipedia but nowhere could I find a heuristic making the conjecture that $e$ is not a period more plausible than its negation. Does anyone here know of such an argument?