Just a couple of thoughts:

IF.....and I mean IF you really have correct injector pulse then you have both proper cam and crank signals (injectors will NOT fire without both), key on voltage at the coil should only last a couple of seconds, it will turn on the asd once it sees crank pulse.

The simple fact that the PCM is cycling the asd relay on "key on" at least indicated that it is powering up

The coil feed (green with orange) is also the injector feed so it you have correct voltage there it eliminates an ASD relay problem

The coil return to the JTEC is gray and will go to connector C1 which is black (sorry I don`t remember the pin number and i`m not where I can look it up but I think it may be pin 7) which is the right most connector looking AT the module (they number C1 thru 3 right to left with them facing you) personally in a perfect world I would use a scan tool to actuate the coil output and see if the module is capable of switching the output...but with lack of a scan tool your only choice at this point is to ohm out the return and make sure it is intact and also not shorted to ground somewhere.

If all that is intact then it is a good possibility that the coil driver failed and it is time for a new "box"


Disclaimer: This all assumes that it really does have correct cam and crank
Rick