Let there be given two identical lumped element resistors $R_1=R_2$ whose heat capacities are also equal and given $C_1=C_2$. We assume the resistors are attached to thermostats, one at temperature $T_1$ and the other at temperature $T_2$ but $T_1 \ne T_2$. Now separate the resistors from their respective thermostats and connect the resistors with a transmission line that has a very low loss (ideally lossless), and such that its metal conductor also has very low thermal conductivity (ideally zero). I know this is a contradiction per Wiedemann-Franz but assume it for the sake of argument. I expect that because of the Nyquist noise emitted from the resistors eventually they will come to a common temperature, and since we assume equal heat capacities, $C_1=C_2$, the common temperature will be $(T_1+T_2)/2$.
Now somewhere along the transmission line whose wave impedance is $Z_0=R_1=R_2$ we place an ideal lossless reactive filter and/or ideal impedance transformer ($I_2=I_1/N, V_2=NV_1$). How will the system equilibrate if not all frequencies are allowed to pass by the filter (e.g., the transformer does not work at $f=0$)? What is the equation that describes the temperature development of each resistor as noise waves are exchanged between them?