Energy balance paradox

Consider a closed system consisting of a room surrounded by walls at temperature $T$ (infinitely big, so $T$ remains fixed). In this room, we place a power source with a wire of a particular material, which has a finite resistance. The power source is kept at temperature $T$ because we glue it on the walls while the wire is left suspending (in vacuum or air).

Initially everything is at the same temperature $T$ and the circuit is open, so no current is flowing. In that case, there is no net energy flux between the wire and the walls of the room, since they are at the same temperature.

Now close the circuit, so as to establish a current in the wire and wait until the steady-state is reached. The wire is going to heat up, because of the Joule heat $I^2R$. There will be a net energy flux from the wire to the walls because $T_text{wire} > T_text{walls}$. But this is fine, after all the power source is a source of energy and its energy is getting dissipated inside the wire and radiated towards the walls, everything is understandable.

However… here comes the problem.

It turns out that if one takes a closer look, whenever there exist a temperature gradient and a current across a single material, aside from the Joule heating, there is also a Thomson heat. This heat can be either negative, positive or null, depending on the sign of $tau vec J cdot nabla T$. It doesn’t matter how small or big the Thomson coefficient $tau$ is, it is always possible to pick a current small enough so that the Thomson heat becomes bigger in magnitude than the Joule heat. It appears that $nabla T$ changes sign around the middle of the wire (but not quite exactly at the middle, due to the Thomson heat) and so about half of the wire will have a Joule + positive Thomson heat contribution while the other half will have a Joule + negative Thomson heat contribution. It turns out that $tau$ is temperature dependent and if we carefully pick a material whose Thomson coefficient is greater at lower temperatures (i.e. if $tau$ decreases when $T$ increases in the range of temperature $T_text{walls}< text{temperature} < T_text{max}$) then the net Thomson contribution will be a cooling, and that this cooling can be greater in magnitude than the heating due to Joule + positive Thomson heats. In other words the wire overall cools down more than it heats. In that case the total net energy flux between the wire and the walls will be the other way around compared than to the previous case, i.e. now the walls are heating up the wire. In this case, the power source has to spend energy (the power is worth $P=VI$) and the walls are also spending energy to heat up the wire, but the wire is left at a temperature that is below that of the walls.

What is going on? Where is the energy going? It’s as if the power source spent energy cooling the wire, the walls spend energy heating the wire and overall the temperature of the wire is, in average, lesser than that of the walls. Is that possible? If not, where is the error in the reasoning? To me it looks like possible, after all this would be similar to a refrigerator where one spends X amount of energy to dump Y>X amount of energy from the cold source to the hot source.