  This is the second of a series of three papers examining how viable it is for
entanglement to be sustained at high temperatures for quantum systems in
thermal equilibrium (Case A), in nonequilibrium (Case B) and in nonequilibrium
steady state conditions (Case C). The system we analyze here consists of two
coupled quantum harmonic oscillators each interacting with its own bath
described by a scalar field, set at temperatures $T_1 > T_2$. For
\textit{constant bilinear inter-oscillator coupling} studied here (Case C1)
owing to the Gaussian nature, the problem can be solved exactly at arbitrary
temperatures even for strong coupling. We find that the valid entanglement
criterion in general is not a function of the bath temperature difference, in
contrast to thermal transport in the same NESS setting [1]. Thus lowering the
temperature of one of the thermal baths does not necessarily help to safeguard
the entanglement between the oscillators. Indeed, quantum entanglement will
disappear if any one of the thermal baths has a temperature higher than the
critical temperature $T_c$. With the Langevin equations derived we give a full
display of how entanglement dynamics in this system depends on $T_{1}$, $T_{2}$
, the inter-oscillator coupling and the system-bath coupling strengths. For
weak oscillator-bath coupling the critical temperature $T_c$ is about the order
of the inverse oscillator frequency, but for strong oscillator-bath coupling it
will depend on the bath cutoff frequency. We conclude that in most realistic
circumstances, for bosonic systems in NESS with constant bilinear coupling,
`hot entanglement' is largely a fiction. In Paper III we will examine the case
(C2) of \textit{time-dependent driven coupling } which contains the parametric
pumping type described in [2] wherein entanglement was first shown to sustain
at high temperatures.
