The heat would always prefer to go one way, but in the reverse direction it would be slower
So if I put that material between two compartments that experience uncorrelated temperature fluctuations, one of them would heat up? And if that works at small enough scales, wouldn't this be Maxwell's demon? I never learned a lot of thermodynamics, so what does this mean for the second law?

h-BN is a good conductor of heat
@sigh
IMHO -my guess is that it would be used as part of the substrate to facilitate heat dispersal ...
then there is the last paragraph
This type of 3-D thermal-management system can open up opportunities for thermal switches, or thermal rectifiers, where the heat flowing in one direction can be different than the reverse direction," Shahsavari said. "This can be done by changing the shape of the material, or changing its mass – say one side is heavier than the other – to create a switch. The heat would always prefer to go one way, but in the reverse direction it would be slower
not maxwell's demon so much as a new switch/rectifier

i am sure Thermodynamics or Antialias_physorg could expound upon this with more clarity

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So if I put that material between two compartments that experience uncorrelated temperature fluctuations, one of them would heat up?

Not infinitely so. Transport will still be along the gradient from hotter to colder. It just means that when area A is hotter transport towards area B will be slower compared to heat transfer towards area A when area B is hotter.
With truly random heat fluctuations on both sides you would end up with a mean heat gradient. It will saturate because the chance of getting a "hotter than current" temperature from your temperature fluctuations will diminish as that end gets warmer (i.e. the chance the ambient temperature cooling it down will increase. Conveserly for the 'cold' end the chance of ambient temperature being higher will increase)