Apologies if this has been answered already, but I checked the FAQ and these forums and found nothing, so here it goes.
In short, does this combo work?
I can see an argument both ways.
The argument for "yes" is fairly straight forward as that seems to be what the cards are meant to do and, if I had to guess, appears to be their intent as well.
But, if we look at it with very strict interpretation of the wording, the argument can be made that Emergency Shutdown CANNOT be used if the only successful run on HQ in a turn was one initiated by Sneakdoor Beta. I say this because the run intiated by Sneakdoor Beta is on Archives and then TREATED like a run on HQ, but it isn't actually a run on HQ.
While being "treated" as a run on HQ seems like a fairly obvious indicator that Emergency Shutdown could be used, it isn't clear. Also, in the FAQ, this is written in regards to Sneakdoor Beta, "Yes. The run is initiated under a specific condition: that if successful it is treated as a successful run on
HQ. This effect lasts until the end of the run, regardless of whether or not Sneakdoor Beta is trashed."
This could indicate that the treatment of the run as a successful run on HQ ends with the run, therefore Emergency Shutdown could not be used later in the turn.