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The authors argue:
Nitrogen execution forces prisoners to participate in their own death, which they consider cruel
Delivery by mask is unproven and could lead to problems (like CO2 buildup).
the protocol is heavily redacted and many other executions have been botched before, which does not inspire confidence.
a lack of oxygen can lead to convulsions, which can prolong the execution if the airways are affected
in case of a failed execution the prisoner is entitled to medical help. This could be difficult or even dangerous to administer in an environment of little to no oxygen
mice did show a fear response when executed with nitrogen
While I do agree with those points and oppose lethal punishment myself, I would not expect the arguments to make a big legal difference. All of them do apply to other execution methods as well, and usually much worse. Personally, I would prefer a death by nitrogen to any other method on offer, if there is to be an execution.