  A wide class of $1+1$ dimensional unitary conformal field theories allows for
an explicit construction of nonequilibrium "profile states" interpolating
smoothly between different equilibria on the left and on the right. It has been
recently established that the generating function for the full counting
statistics of energy transfers in such states may be expressed in terms of the
solution to a non-local Riemann-Hilbert problem. Following earlier works on the
statistics of energy transfers, in particular the ones of Bernard-Doyon on the
"partitioning protocol" in conformal field theory, the full counting statistics
of energy transfers in the profile states was conjectured to satisfy a large
deviation principle in the limit of long transfer-times. The present paper
establishes rigorously this conjecture by carrying out the long-time asymptotic
analysis of the underlying non-local Riemann-Hilbert problem.
