I searched but did not find much on this topic (other than posts that all seem to indicate why you should keep the portafilter IN the group to keep it up to temperature - i.e. so you do NOT cool the group when it is inserted.)

But, having an Elektra A3 with a quickly heating grouphead, it would seem to me that by inserting a COOL portafilter AFTER the cooling flush, you would be helping OFFSET the rising temperature of the group.

In my machine, I do notice that over the 25 seconds or so it takes my shot, the water temperature INCREASES... mostly towards the end of the shot.

Has there been any repeatable temperature tests showing hot vs. cool portafilters in an HX?

The cool PF will suck heat from the group, but it will happen pretty slowly and not to a large degree. The real concern would be the cooling effect on the espresso itself via the PF bottom and spouts. It would make sense with a naked PF, still not terribly practical though.

It would also detract from easy repeatability. You'd have to wait for the portafilter to cool between shots.
Just have a queue of portafilters for multiple shots and you're set for that next party....

Okay, you could cool in the sink, but still seems more trouble than other means of stability...

Have you tried other, more effective methods for reducing group temperature yet? For example, with an exposed group a wet towel will make a significant difference. Setting the pressure lower in the boiler will also make a noticeable difference.

I just tried putting a cold portafilter into my 15 pounds of brass lever group and it took two and a half minutes for the portafilter to feel warm near the ears, with the bottom still feeling cool. Given the small contact area on the ears, it isn't surprising it took so long. So, while it will reduce the temperature of the group, I have doubts about how quickly it will work. I also worry about the effect of the espresso hitting the cold bottom of the portafilter, I know pulling a shot into a cold cup has a negative effect.