[embee]

Back in the dim and distant past I was on a group where we assessed external technical submissions to the company ("forward engineering"). We got a lot of things like this to consider.
The fundamental questions are what is the thermodynamic cycle, the inherent efficiency limitations, and as said above, what is the energy source.
This is a typical thermodynamic compression/heat input/expansion cycle, and so the thermal efficiency is readily evaluated. The rest of the device is just mechanism wrapped round the cycle to get the work out in a usable form.
Distilling devices down like this meant that nearly all such "novel" ideas were just a standard process in a slightly different physical shape wrapped up in different nuts and bolts.
Thermodynamically I'd be very surprised if it is any different to any other cycle either in common use or having been used at some time. Mechanically it's questionable why a linear arrangement should be much more practical or desirable than a rotary device (air springs are not 100% efficient for example, heat is lost). Component/material temperature regulation is frequently an afterthought in these things.
As far as their claims for emission/efficiencies, it really does come down to what the chemical process is which provides the heat energy, and I'd want to see a lot more detail. In principle thermodynamic cycles are more efficient the higher the temperature and pressure they operate at and the shorter the cycle times (heat loss), so their claims for a low temp/press "reaction" process would be a big question mark in my mind.
Very often tech submissions we got to assess ended up with some element of "perpetual motion machine" included in order to achieve the sort of efficiencies being claimed.
The reasons why the typical internal combustion engine still works more or less as it did 100yrs ago are fundamental, you can't cheat thermodynamics. Mechanisms and materials evolve, but the process stays the same. It is not for the lack of ideas.