These engines use valve ports, so piston motion produces the valve action. All the timing is set by the engine geometry. There are few variables adjustable at run time. This is also true of Wankel engines, which is why Wankels hit a wall - you can't tweak fuel injection timing or valve timing. In a Diesel, you can't even tweak the spark timing. Much of the improvement in IC engines has come from active control of fuel injection and spark, along with a few attempts at dynamically tweaking valve timing. These designs don't seem to be able to do that.
"and allows the engine to run with diesel-like efficiency and power, while maintaining gasoline's emissions benefits."
Whether or not any of this is actually feasible is explored in elbigbad's sibling comment.
https://en.wikipedia.org/wiki/Dark-class_fast_patrol_boat
Dark Avenger probably being my favourite!
https://upload.wikimedia.org/wikipedia/commons/a/a6/John_Eve...
2 strokes are terrible engines generally, because they have very high emissions, usually because they release unburned fuel in the exhaust.
Gasoline is comprised of a shorter set of hydrocarbons (mostly 6, 8, and 10 chain) than diesel (about 12-16 chain length, though 8-21 carbon are listed at Wikipedia). Volatility (and ignitability) increase with shorter chains -- methane (C2H6) and butane (C4H10) -- that is, two and four chain hydrocarbons -- are both gaseous at or near room temperature.
Longer chains still get you heavy fuel oils, tars, etc.
...and eventually that results in simple plastics like polyethylene, which is basically a very long chain hydrocarbon:
We already have this technology. Diesel engines ignite the fuel through compression. We also had the opposing pistons layout for a while (https://en.wikipedia.org/wiki/Opposed-piston_engine).
EDIT: we even have triangular-y shaped opposing engines (https://en.wikipedia.org/wiki/Napier_Deltic)