It's possible that there is a cut-off switch in the out socket, but it's unlikely. The device is completely passive, there would be no reason for the manufacturer to include a switch.
Anyway the the picture you show seems to be of a mono socket, the device has "stereo" sockets (actually a balanced line outputs) probably more like this one (if indeed it does not have a switch):
You are using an unbalanced signal, coming from your stereo, that you connect by means of the mono jacks on (each of the channels of) the input side. When you insert the mono jacks you are actually short circuiting the ring and sleeve (1 and 3 in the picture) of the input socket. That is fine, being an unbalanced signal, the proper way to use a balanced line is to short-circuit the "would be" minus to ground.
Now, the Nano patch connects passively "pin by pin" the inputs to the outputs, so this means that in the output side, if all is well, the ring will also be connected to ground.
This is a requirement to get a proper (unbalanced) signal on your monitors. The monitor will subtract the signals from pins 2 and 3 in the XLR connector (respectively, tip and sleeve of the jack; the legend of the figure above is inconsistent with the pin numbering of XLR connectors!)
If the ring of the jack is not grounded, the monitor will have no reference signal to subtract from the actual signal and will produce a weaker signal, which is consistent with your description of the problem.
Anyway, my advice is that you try first to clean the interior of the socket (try a compressed air can if you have one or just blow inside the hole and/or use a small pointed brush). Failing that, I thinks it's worth to open the box and try to see what's happening. After all it's a completely passive device, you may be able to detect visually the problem (if the connector is not enclosed, which may be the case in a high grade appliance).
Consider that if some day you want to use the device with a balanced source signal (like coming from a professional or semiprofessional mixing board), then you may not be able to use that channel, even with the recessed jack...