More starting problems

I recently purchased a 1986 supra automatic(!) which has a starting problem that no one can seem to fix. If the Supra sits for a while, (an hour or more) she starts fine and runs great. If she runs long enough to warm up, and I turn off the ignition, she won't start right away -the battery seems to be sending power to the starter, but the power is not getting there. I have checked the battery and it shows power to spare, I have had the starter & alternater replaced, the Garage has rewired fron the battery to the starter and checked the relays(which seem okay), I have even had a new ignition module installed, but the starting problem still persists. What makes it really frustrating, is the problem seem to happen randomly, as the garage will run the car to warm up, the it won't always fail, so they can't diagnose the problem. If anyone has any suggestions, I would really appreciate the feedback.

my 87 does the same thing....did you ever get any remedies????

Your egr's are to high. Replacement of your egr valve with a throttle body cleaning should fix it.