The problem I see is that existing engines are already good enough and cheap, so it could be a hard sell for manufacturers.
The "an alarming amount of fuel goes out the tailpipe" is reminiscent of the wording associated with various fuel-saver scam gadgets. In practice, this is measured in HC emissions, which are usually in the low tens of PPM range on a properly tuned engine.
Disregarding mechanical sources, engine noise is largely caused by the high-pressure gases flowing out the exhaust with pulsations on each power stroke. The distinctive beating of a low-cylinder-count engine e.g. a motorcycle is entirely this phenomenon. To attribute the noise of a properly functioning (i.e. not backfiring) engine to "unburned fuel" is completely wrong.
The Liquid rotary fires thrice per revolution, or six times faster. The go-kart engine is designed for 10,000 RPM. That puts its fundamental frequency at a less objectionable 500 Hz.
But the go-kart video did not sound like 500 Hz -- it sounded like about 100 Hz.
As for the "alarming amount of fuel out the tailpipe" line, maybe they're comparing a two-stroke engine. I don't really know how unburned fuel affects a two-stroke's sound.
http://liquidpiston.com/technology/x-mini-gasoline/
If these are indeed quieter, then I could see them commanding a significant premium for lawn tools, generators, etc.
There are a lot of customers there these days.
The Wankel's design limits the eccentricity of the crank shaft which consequently limits torque to relatively low levels compared to a contemporary piston engine of similar fuel consumption. There is no torque advantage to a Wankel, it's at a disadvantage.
The "secret sauce" that the Liquid Piston folks bring to the party is the upgraded rotor shape. It doesn't need apex seals like the old Wankel version did. The news here is that they had reduced their experimental work into an actual functional engine. Unlike the Wankel, their cylinder design would really support a compact multi-rotor engine but presumably you could gang them together on the same crankshaft if you really wanted to.
Other than the UAV and possibly the paramotor market I'm not sure where these guys are going to find customers though.