keep adding loose nut , and try to turn it out ,
it took GDE 5 days to remove 1 glow plug ,
the design is to blame , your story 's only good part is that you say you can move it ,
with no head to grab on to it will make it harder , you can only hope you were almost there .
if you counted the turns you had , you can evaluate your progression from
the images , pushing crap out of the top threads were it accumulates.
going from pictures , with hex head gone , I guess you do not have a lot to grab on to ....
---------------------
Going by images ==> 6 complete turns before the threads start to make their way out...
...
this thing is threaded in , I don't see how starting the engine would push it out ....
....
some had to remove cylinder head to deal with this ....
..

 

The problem just may be that gap above the threaded part of the plug in the head. Debris gets in there, deposits harden and as you back it out it balls up on the threads and ...kaput. It's a bad design that has caused this problem and easily seen in the picture above.

If there is still thread visible, dump a bunch of cleaner on it and try blowing air on that cleaner to hopefully get rid of debris and maybe lubricate things.

 

Can you expand your thought on "keep adding a loose nut" please? if i could find a reverse thread 5or 6mm nut that would be ideal., but getting that sucker on and started would be a complete pain without removing the HPFR.

At least by looking at the picture it sure looks like it wont blow out causing more damage....I had about 3 turns, by your math its half way out, or just past the head threads, which explains the ease of removal i felt. That's where I made my mistake, going from 1/4 out 1/8 in to 1/2 or better out, then in. I though because i cleared the threads it was easy street from there, but it still popped. For those who haven;t but are contemplating this, remain vigilant with the 1/4 in out rule, and don't be hasty. Even if it seems too easy take your time!!!

 

there was a '' recipe '' posted of different loose nut products sprayed in there ,
it makes its way over hours and disolves , loosens , the crud in the threads ,
it was a tuff job , ,,,,,,,, now with the hex head gone ,,,,,,,, ouch .
you could try calling GDE , they may suggest different product and tell you their opinion
on your particular predicament ...
+1 248-977-9531 or support@greendieselengineering.com

 

Try to cut a slot across the glow plug shaft using a Dremel if enough room, Basically looking like a slotted screw head then use a hammer impact driver make sure the impact tip and slot are very tight, the hammer action along with twisting action will probably work..

Good Luck and I’d get the engine good and hot if possible.... Let us know what’s the final results....