Poor Mark. He's on the phone (my phone) with Samsung, even as I type. We've narrowed down the source of the problem to being Google pushing out the 5.1.1 update. Mark was only on version 5.0.2, so his phone was due for an update, which started being rolled out for Galaxy S6 yesterday.
The lady on the other end of the line walked him through several different steps, none of which were effective, to try and revive the phone. She finally recommended that we visit a Best Buy, as they are an authorized "Samsung Experience" center. Surely there's at least one Best Buy in this area, so that shouldn't be hard to do.