I'm really excited to see progress being made in this area! I haven't read the paper but looked into this area about 10 years ago, and the two questions that jump to top of mind are:

1) What thermal transfer mechanism being controlled?

2) What is the dynamic range in terms of conductance? (Looks like 13x)

There are a number of different mechanisms for energy transfer at this level, and it seemed like the most straightforward way to build a thermal transistor was to get a material that was an insulator for all but 1 transfer mechanism (usually phonons) and then use an electric field to try to permit/block that. The solutions that did exist back then seemed to be very leaky, with thermal transfer while "on" was only about 3x that of when it was off, and this was for microscopic structures where it wasn't obvious how they would physically scale to useful sizes.

I would _love_ a practical thermal transistor technology for an energy management system in a passive house. Allowing your house to "breathe" at night during the summer without things like loud attic fans would be fantastic.