  Exchange energy statistics between two bodies at different thermal
equilibrium obey the Jarzynski-W\'ojcik fluctuation theorem. The corresponding
energy scale factor is the difference of the inverse temperatures associated to
the bodies at equilibrium. In this work, we consider a dissipative quantum
dynamics leading the quantum system towards a, possibly non-thermal, asymptotic
state. To generalize the Jarzynski-W\'ojcik theorem to non-thermal states, we
identify a sufficient condition ${\cal I}$ for the existence of an energy scale
factor $\eta^{*}$ that is unique, finite and time-independent, such that the
characteristic function of the exchange energy distribution becomes identically
equal to $1$ for any time. This $\eta^*$ plays the role of the difference of
inverse temperatures. We discuss the physical interpretation of the condition
${\cal I}$, showing that it amounts to an almost complete memory loss of the
initial state. The robustness of our results against quantifiable deviations
from the validity of ${\cal I}$ is evaluated by experimental studies on a
single nitrogen-vacancy center subjected to a sequence of laser pulses and
dissipation.
