The light minute has everything to do with being able to conclude are out of eachothers lightcone. The only frame where the boxes would be opened simultaneously is the frame the clocks were synched in.
What significance does one minute bring to the thought experiment other than remove the possibility within SR that event E caused event F (i.e., event E couldn't cause event F since event is space-like separated and there is no causal influence at the time the boxes are opened)? The same experiment could be performed at distances much closer (e.g., earth's orbit), but then you have causal relations that, within the scope of SR, operationally labels the events as simultaneous.
You don't need to go through all that to conclude the reference frames changed when the clocks were separated.
I think you do. The reference frames don't necessarily change because box F travels away from box E (otherwise nothing in the world would be in synchrousness since everything is in motion, including the continents in case of continental drift, earth, solar system, Milky Way, etc. What absolutely affects reference frame changes in SR (correct me where I'm wrong) is how you travelled (i.e., relativistic velocities change reference frames), or how far you travelled (i.e., did you move outside the light cone such that a casual chain cannot be operationally defined). Note: I keep mentioning the term 'operational' in regards to SR because this is how, I think, Einstein treated the notion of synchronous clocks in SR. If you cannot show that the clocks are synchronous in some operational manner, then you do not possess a concept of simultaneity. If such a operationalist method cannot be done (e.g., due to relativistic velocities or travelling too far outside the light cone such that there are no causal chains connecting two events), then the concept of simultaneity is lost. What travelling one light minute does is firmly establish that the opening of box E cannot be causally tied to opening box F, and therefore from the operationalist perspective of SR, both clocks on box E and box F are not known to be synchronous at the time of opening of either box. Hence, in terms of the quantum probability you must allow for the 3 states that Barrett mentioned.
Since when does SR [or GR] pretend to know how events are ordered behind a horizon anyway?
I'm not sure what you mean. Since Einstein within SR (versus GR) defined simultaneity in operationalist terms, the theory, I believe, can only determine if two items are synchronous if operationally it is possible. What, I believe, Einstein showed in SR, is why it is not possible to establish absolute simultaneity in nature since any attempt to do so in an operational manner would not be possible [i.e., in circumstances dealing with certain relativistic situations if you accept the postulates of special relativity (which experiments indicate that we should accept - at least from all experiments conducted thus far].
You are claiming that QM and SR are logically inconsistent based on the thought experiment posted. Let a signal be sent confirming when Fred opened his box and allow enough time for it to reach Elle and then they can determine who opened their box first. One thing is for sure the answer won't conflict with the prediction relativity would make for this scenerio.
First off, I wouldn't be so presumptous to say that special relativity can be invalidated using this kind of thought experiment. The issue for me is purely a philosophical manner as to the issue of realism and not the invalidation of one of the best scientific models of science.
Let a signal be sent confirming when Fred opened his box and allow enough time for it to reach Elle and then they can determine who opened their box first.
The thought experiment was specifically designed to conform to Einstein's operationalist method of clock synchronization. That is, if Fred sent a signal to Elle at 12:01 after opening his box with the message "I opened my box at 12:01", then once Elle got the message she would know that Fred's clock said 12:01, but since this cannot be operationally defined in terms of determining if their clocks are still synchronous (i.e., Fred is now outside Elle's reference frame such that their clocks cannot be shown to be synchronous), therefore how can this signal from Fred settle the issue?