That's the same question that comes up in the original transparent-boxes problem, of course. We probably shouldn't try to recap that whole debate in the middle of this thread. :)

Don't worry; I don't want to do that :). If I recall the original transparent-boxes problem correctly, I agree with you on what to do in that case.

Just to check my memory, in the original problem, there are two transparent boxes, A and B. You see that A contains $1M and B contains $1000. You know that B necessarily contains $1000, but A would have contained $1M iff it were the case that you will decide to take only A. Otherwise, A would have been empty. The conclusion (with which I agree) is that you should take only A. Is that right? (If I'm misremembering something crucial, is there a link to the full description online?) [ETA: I see that you added a description to your post. My recollection above seems to be consistent with your description.]

In the original problem, if we use the "many choosers" heuristic, there are no choosers who two-box and yet who get the $1M. Therefore, you cannot "choose to be" one of them. This is why two-boxing should have no appeal to you.

In contrast, in your new problem, there are two-boxers who get the $1M and who get their E module to output True. So you can "choose to be" one of them, no? And since they're the biggest winners, that's what you should do, isn't it?