Okay, a Quick recap of the path of power to cause a score reel to reset towards zero on start up:
After the Start Relay (S) has been energized, it pulls in the AX relay armature to latch with a pulse through score motor switch 2C.
AX contains a switch to keep the score motor running since the S relay will drop out after the first score motor cycle is completed.
With the Score motor running, a pulse is continuously generated by a score motor switch at 1A (BLK-WH and Red-Wh). Since the other score reels are resetting, this part of the circuit is fine so far. This includes any connection through the Jones plugs up to the head.
Then, there are 4 individual switches on the AX relay which route those pulses through to the score reel assemblies. This takes the place of the Player unit switches called out in Steve Fury's reset animation linked to above. They all share a Blk-WH wire (or solid jumper). The other wire colors are as follows:
10's: Gr-Red
100's: Br-Red
1000's: Or-Red
10,000: Bl-Red
The above wires then go to the individual score reels to their "run out" switches. These are the one that are supposed to be normally closed at every position except for 0. Not the switch that closes at 9 and not the switch that is closed only at 0.
The run out switch is directly wired to the score reel step up solenoid.
So, with the condition you currently get (Reset sequence starts but two score reels don't advance while two others do), get out that trusty Alligator Jumper lead and first jumper across the switch on the AX relay (see wire colors above) and see if one of the dead score reels begins to pulse. If it does not, then move your jumper to the run out switch called out above (same wire colors as the AX relay switches). They actually share the same wire color on both sides of the switch with the one going directly to the score reel solenoid. While a simple swipe with an ignition file and visual inspection for proper gap is all that is normally required, the jumper will definitively rule out or find a malfunctioning switch in this manner.
With all 4 score reels sitting at zero, a series of 4 normally open "reset" switches (also refered to as zero position switch) should be closed and will complete the circuit to trip (relax) the AX relay through a pulse from score motor switch 2B (Or-Wh / Red-Wh). This pulse occurs at the very end of the score motor cycle after the 5 pulses from 1A have been sent to try to step the score reels towards zero. If one or more reels are not at zero (or the switches are not making contact or there is a broken wire or cold solder), the score motor will continue to turn to send more pulses through the AX relay switches. This will go on forever until all four score reels signal they have made it to zero through their reset switches.