Just updated to the latest firmware for the E6 (had to do it manually, though, it wouldn't find it via the online updater).
I'm impressed. The HDR Game Mode seems to be the HDR Standard mode with more options turned off (e.g. RealCinema is forced off, when it was forced on in HDR Standard). Calibrated to match my HDR Standard settings, HDR Game Mode looks pretty much exactly the same. If there are substantial latency differences, I'll be extremely pleased. I was expecting a kind of half-way HDR Game Mode, whereby it never looks quite as good as the "normal" modes due to turning high-cost processing off, but it looks exactly the same to me.
I was "fine" with the latency before (playing Uncharted 4 Multiplayer to test it out, it never felt too "off"), so if it's really improved here it should remove one of the main reasons people steer away from these fantastic OLED sets.
I haven't played around with it too much, and I don't have any way to get concrete proof or anything, but my initial impressions are positive. Placebo or not, I can't be sure, but I can say that the audio-slightly-out-of-sync issue I had in 4K HDR videos is now resolved, at the very least.