So the net effect is that the device in the lab gets cooler and the object far away get hotter. So you have a flux of heat. But if the other object is hotter, you have a lux of heat from in the wrong direction, that is impossible according to the Second Law

I don't think your hypothesis would necessarily break the Second Law. Wouldn't the hotter distant object result in making Brownian motion around it? The same Brownian motion which would eventually end up as energy input at the graphene membrane - provided a thermodynamically closed system?

As so I'm not convinced this study breaks the Second Law, nor do I see how it produces useful work. To me it only seems so, because the system isn't being modeled as a closed system...

EDIT: Actually, I'm pretty sure the following statement is completely false:

the graphene and circuit are at the same temperature and heat does not flow between the two.

I'd suspect heat does flow between the two, it's just outside their modeling of the system.