No, but I think it's been narrowed down somewhat. It could be the solenoid, or "reverse lockout switch" or a loose wire...
I looked in the trouble shooting pages of the manual, and that's about all it can be at this point. I'm taking it in for a tune-up in the morning, so hopefully, they will find it, if they have to turn it on more than once.
It starts just fine, but if I have to start it again within the next 15 minutes, I get nothing.
Geez! Wish I could help, but my mechanical inclinations are less than yours.
I hope the tune-up will do the trick.
I have to take my '95 Camry for inspection this month and hope they find nothing wrong with it. It's got low mileage and is in very good conditions as i don't drive it much. But last year it failed the exhaust diagnostic and cost me 280 bucks. Last month spend $640 on brakes. About time to consider trading it in. :)