HFM and LH managed engines don't have an additional injector or cold-start valve.

Here's an update:

Checked all the sensors and the vacuum lines: all passed within spec.

I decided to check the fuel accummulator (adjacent to fuel filter on
W124) by plugging the rubber fuel-house and disconnecting it from the
accummulator and draining out all the existing gasoline from the
accummulator.

Then I started the car, and found the accummlator leaking (e.g., not
holding pressure).

Now I could do two things:

1) replace the accummulator ($70);
2) by-pass the accummulator (wagon specific pn# 124.470.48.64 $15), like later year cars (1993+) with HFM/LH (my 1988 W124 wagon has an HFM C36 M104 engine in it)

Since the accummulator is only needed with the CIS injected cars, I
decided to by-pass the accummulator by installing the bypass-pipe, new
copper crush-washers, and plugging the rubber fuel-line. I also
replaced the fuel-filter (use 002.477.4401 for tandem pumps; greater volume; and dampening effect) and the fuel-pump check-valves (pn#
000.090.03.10 , $12/each)

BTW: here's a tip diagnosing hard-starts on HFM and LH cars assuming
fuel-pumps are working:

1) hard "cold" starts with long cranks: check accummulator;
2) hard "warm/hot" starts: pull vaccuum line from fuel-pressure regulator and "sniff"/check for gasoline

note: you "sniff" by waving you hand over the "source" and waving the
smell to your nose, DO NOT PLACE YOUR NOSE AT THE SOURCE AND SNIFF
(Chemistry 101).

:-) neil
1988 E36T AMG
1993 500E

The cold start system in the HFM system is the same as the LH system and
the reference manual is the W140 intro.

Cold start additional fuel is supplied only by the normal injectors
being held open for a longer period.

The sensors for cold start include:
The engine coolant temperature sensor
Camshaft rpm from the cam position sensor intake manifold vacuum from the EZL/AKR ignition control unit
and, last but not least, the length of time the engine has been turned off.

Basic fuel quantity is determined by the air mass sensor and the engine rpm.

after start enrichment is controlled via
coolant temp while cranking and gear selector lever position

warm up enrichment:
coolant engine temp at start up
intake air temp
engine speed and load
idle and partial load conditions