All GonG joints have a 10:1 taper ratio if I'm not mistaken, so the measurements at the tapered end doesn't really matter.I was trying to illicit the needed data from @stark1 but no luck.
Knowing that small diameter, you'd know the new stem shape.
Not having a taper slide really held back my lathe for decades. I've put it to good use.
For your reference;
Edit: I just measured mine and it starts at 9.3mm and tapers down to 8.9mm.