If this were a throttle-body injected engine, I'd say it could be gunk in the fuel occasionally clogging the injector. But I believe this is multi-port injected; one clogged injector would not completely kill the engine, and it seems unlikely you'd have multiple injectors clogged just enough to kill idle but not enough to affect off-idle performance.
Some kind of clogged pickup screen in the tank seems possible, but I'd think that would also affect performance when off-idle - which you don't report. It would be nice if you could measure fuel pressure when the event occurs.
A sticky IAC valve, or intermittent MAF sensor connection fit the intermittent failure, but don't explain the correlation to refueling. You have only two data points. With 15-20 I'd be more confident - how about running a longer experiment?

One issue I know you have is too many engineers and not enough technicians.
