1998 Chrysler Town and Country Random Starting Failures

I have been trying to find someone with an answer for a totally random starting problem with my "˜98 Chrysler Town & Country LXI, 3.8L for over a year now. Regardless of the variables I’ve listed below, everyone I’ve taken it to has been stymied by the van’s inexplicable and unpredictable faulty starts.
I have noticed that the only constant with this, is that I can get the engine to willfully turn over exactly 6 times without hesitation. It will run for about 2 seconds, then shut down and on precisely the seventh try it does absolutely nothing. It doesn't matter if the weather is hot or cold, wet or dry. It doesn't matter if I've just turned the van off, or if it's been sitting all day and night. I've attempted to lock and unlock the vehicle various ways, being suspicious of an issue with the security system, but even when I left it unlocked, I've had this problem. It may behave fine for several weeks, even months without recurring or it may happen again within 2 days of the prior occurrence. Once I've reached that seventh attempt and it has no response, it may start again in a few minutes or up to a few days later. It's totally random in occurrence and recovery. The only thing for sure is that it will do a complete start 6 times, shut off in about 2 seconds and not even respond to the ignition on the 7th time.
I've been to 4 shops, including our area's preeminent electronics expert. No error codes are logged. I've left it for almost 2 weeks at the Chrysler dealership to have them attempt to recreate this problem, but of course, it wouldn't fail while they had it! I've waited for up to eight hours to attempt to start it again between the first six starts, but it seems that it will not go to the final total failure unless I've actually completed all six starts and failures, regardless of how far apart I space them. I've even disconnected the battery cable for various lengths of time between attempts, but it still "remembers" how many tries I've made since the initial false start, before failing to start entirely. It appears it will not recover until it fails entirely on the 7th try, then it’s a crapshoot at when it will finally turnover again.
I haven't had a consistent problem with the instrument panel failure at startup, although, I have had the panel dials flat line and the odometer/trip meter displays shut off, while other lights (ABS, Service Engine Soon, Seatbelt, etc. ) Begin popping on. This occurs mostly as I travel down the road and resets itself randomly, sometimes minutes later, sometimes days later. It doesn’t affect the overhead display for mileage, temperature, direction, etc. I’ve tested shutting the vehicle off to reset the displays, but the situation seems to also recur and reset just as randomly. I’m not sure that they are related issues, but I’ve included this information just in case.
Is there anyone who’s had experience with this or can share some insights?
Thank you!