The reason the engine starts and then stalls is because the fifth injector is doing its job, but it only runs for a few seconds, so we know it is ok. It sounds as if the mixture is too lean when cold, and there one or two reasons for that.
1) The control pressure regulator (CPR, aka warm up regulator) is not working.
2) The airflow sensor plate is sticking in its plenum camber
I would rule out the auxilliary air valve ( A-A-V), as you say that any attempt to accelerate the engine leads to it cuting, and the A-A-V has the effect of holding the throttle slightly open.
Take the large round rubber duct off the fuel metering unit (FMU) and check with a feeler gauge at 120 deg intervals around the circumference of the disc that the gap is even. Check for smooth operation of arm to which the disc is attached by lifting it up. It should not have any trace of stickiness.
If there is no fault found here then the the most likely cause is the CPR.
In simple terms, the upward action of the sensor plate and hence the angle of the arm, is opposed by the system pressure, and the amount of fuel metered is dependent on the angle of the arm, because as the arm moves, it draws a plunger up into a ported chamber through a waisted (not wasted) section, which then controls the fuel flow to the injectors. 924 injectors do not open and close when the engine runs. Fuel forms a pool behind the inlet valve whilst it is shut, and when it opens, spray from the injector re-atomises the fuel.
It follows that if the system pressure is lowered the arm moves up more and meters more fuel for a given air-flow. The job of the CPR is to open when the engine is cold and by-pass the FMU allowing the system pressure to drop slightly.
The regulator on your car may have failed in the open position at sometime, and subsequently, the over-rich mixture has been reset at the FMU leading to you having a mixture which is too lean when cold, or there may have been a loss of electrical supply to the regulator giving rise to the same symptoms, and remedy.
Alternatvely the regulator may have failed (i.e. stuck) in the closed position, in which case no mixture enrichment will take place.
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