PBA,
I am saying that it would cause a Chillul Hashem, in sense that the world would say the opposite of ??? ????? ???????? ??????? ????????? ??????, but that isn’t the reason that executing would be assur.
The reason executing would be assur is because there is no Sanhedrin sitting in the lishkas hagazis, and the potential exception of Chillul Hashem certainly doesn’t apply.
If I could edit it, I would go back and insert the words “if anything” (offset by commas) before the words “would cause” in the last sentence of that post.