I understand your Problem and I also think we could explain this a bit better. So I will try that
Since the pre-chamber is mostly likely fed by the main-chamber charge without it's own injector, it's important to not only achieve a good "cleaning" of the pre-chamber during the exhaust stroke, but also controlling the fuel concentration of the part of the mixture which goes into it.
To feed the pre-chamber precisely, you need to control the fuel injection timing to the right position of the piston and most impportantly to the right movements in the main chamber. This heavily depends not only on intake Timing,but also to the intake port shape. If you have a side-mounted injector, the goal would be to create a heavy charging motion with a tendency to create a big swirl counterwise (if exhaust is right and intake is left). This is exactly achieved by having a "mountain"-type piston and a very high angle of the intake port.
If you achieve this, you are able to allow a charge motion into the pre-chamber. This can be intesified with a asymmetric design of the pre-chamber. Now for the mixture part. If you have your tumble motion you need to time the injection properly to allow the injector to almost directly inject the fuel into the chamber with one fuel jet, or you decide to inject the fuel primarily into the main chamber ant time it correctly to allow a richer mixture in the pre-chamber. The first option is simpler for operation but harder to construct because you need to have a "extravagant" Sport for the injector, the later is harder to controlm but way simpler to contruct.
The second Problem is the "cleaning". This heavily depends on the timing of the exhaust valves and the overlap, but also very strongly on the shape of the pre-chamber, because it (in cooperation with the main chamber design) has to allow the charge to leave the chamber properly without getting into much trouble with the main charge motion.
Now, if you think these are all just simple expectations from me: They're not. This is based upon the simulations I've made with my own project. If you have interest, I could tell you a bit more about the charge in multiple photos in the next days, but for know I have one picture, where you can see how the ports, the main- and the pre-chamber look like.
If you wonder how this engine will work: Using direct injection and trying to directly feed the pre-chamber during the compression stroke, the goal is to achieve a slightly richer mixture in the pre-chamber (which makes up about 22% of the volume at TDC) and a slightly lean mixture in the main-chamber to average on a stochiometric level. The idea is a combustion, which always firstly occurs in the pre-chamber to auto-ignite the outside mixture in a form of HCCI.
The pre-chamber mixture is the "pressure booster" in the ignition cycle and allows to combine full control with a high efficiency. The pre-chamber mixture can be auto-ignited after the HCCI-principle if the engine is in the right operation engine (which get's increased by the possibility of a richer mixture) of with a spark plug.
To allow a high-efficiency combustion in the pre-chamber even with spark-plug operation, the pre-chamber is designed asymmetrically to achieve extremely high levels of turbulence which allows for very fast combustion.
Now, here a picture of the chamber design with the piston at BDC.
