It was actually the day after (June 16). While it's difficult to pinpoint the exact second we passed, it was between 6:55 and 6:56 (I'd guess about 6:55:45) UTC. The data for W563 shows they were slowly climbing through 29,100 at the time, and we were level at 30,000. Using Google Maps, it appears we were laterally separated by around 2,900 ft. laterally in nearly parallel (but opposite) headings. So I was about right on estimating our vertical separation but we were nearly twice as far horizontally than I'd estimated.

Still, this does verify that we were indeed less than 1,000 ft. vertical and significantly less than 1nm laterally separated. So definitely less than "conventional" radar separation and less than you'd want during IFR, but it was during VFR flight in perfectly stable air, so probably acceptable given the conditions. Absolutely a close pass and outstanding (although very brief) sighting of a Mahan Air A340 in flight. Very cool.