  It has been recently noted that the diffeomorphism covariance of a Chiral
Conformal QFT in the vacuum sector automatically ensures M\"obius covariance in
all charged sectors. In this article it is shown that the diffeomorphism
covariance and the positivity of the energy in the vacuum sector even ensure
the positivity of the energy in the charged sectors.
  The main observation of this paper is that the positivity of the energy -- at
least in case of a Chiral Conformal QFT -- is a local concept: it is related to
the fact that the energy density, when smeared with some local nonnegative test
functions, remains bounded from below (with the bound depending on the test
function).
  The presented proof relies in an essential way on recently developed methods
concerning the smearing of the stress-energy tensor on nonsmooth functions.
