We have a 2001 Sienna that does this *exact* same thing. Had all these fixes tried - throttle body cleaning, etc. to absolutely no effect.
we were told that ethanol in 89 gas might be the culprit, but that was all the dealership had to offer us. It did seem to help for a few months, but now the vehicle is doing it with 87 gas in winter! {Although I suspect ethanol is being used in 87 gas now}.
After all these pricey sophisticated fixes, I'm going with only putting the gas cap back on gingerly, such that it is only finger-tightened, long before the 'click' occurs.
As the problem for us is worse on a 1/4 tank than when the tank is full, I'm wondering if excess pressure is building up in the system and the gas cap doesn't properly release the pressure. This makes sense for Ethanol versus MTBE in 87 gas due to greater vapor pressure of ethanol (ethanol also has a great affinity for water, putting that into your fuel system if any water vapor is in your tank). Of course, don't tell that to the farmers and legislators.
Now we just have to see how much fuel I lose to evaporation once the summer comes into full bloom. So far, car has started fine with loosened cap! {but its only been two days}.
Good luck to those with this problem that the dealerships are stumped by.