This is a dilemma.

On the merits of nitrogen hypoxia as a method of execution:
Nitrogen hypoxia is one of the most peaceful, least painful deaths available, and the concerns of a tortuous death are pretty unrealistic.

The air we breathe is 80% nitrogen 20% oxygen, so the body has no adverse reaction to nitrogen. Remove the oxygen, so the person breathes 100% nitrogen 0% oxygen, and the air will feel normal, there is no feeling of suffocation or shortness of breath. However without oxygen one will lose consciousness within a minute or two, and be dead in 5-10 minutes.

For reference, airplanes are pressurized at high altitudes because the less dense air contains fewer molecules of oxygen per lung volume. Past about 11,000' above sea level, there's not enough oxygen in the air to sustain full consciousness. Here's a video of that- the alarm is going off in that guy's airplane because the pressurization system failed. But in an oxygen-deprived state, he happily reports to the controller with a smile that he's totally unable to control his aircraft but other than that everything is peachy. You'll note he is totally unbothered by his condition.
When the controller orders him to descend to 11,000 feet, air density increases, his brain starts working normally again, and he starts making coherent radio calls.

The point of this isn't to be funny, it's to illustrate that in a hypoxic state he was totally calm and happy and not in any distress at all, even though his aircraft was out of control. So if anything, nitrogen hypoxia might be the most peaceful way to die, as in their final moments the condemned may be less concerned about the fact that they are being executed.

The only possible 'botch' I can imagine, is if either the condemned isn't breathing 100% nitrogen, or the nitrogen is shut off before breathing stops, that could leave the condemned in a state of hypoxic brain damage. That could leave him a vegetable, or alive and awake but brain damaged (low IQ, cognitive problems, etc). That's the sort of state most places consider 'unfit to stand trial' and he gets remanded to a care facility probably for life. And that would require a pretty bad botching to create that situation.

But I still hope the complaint stalls things:
On the other hand, I think execution is a barbaric punishment, and I think we should do all we can to abolish it anywhere it still exists. So I support this group, even though their concerns are unscientific to the point of ridicule.

I also suspect I'm very much not alone here. I'm not a doctor, but I am a private pilot and a scuba diver, so I understand what a body needs in terms of breathing gas a bit more than average. I know for a fact I could write a totally bulletproof execution protocol that would provide a reliable, quick, humane, pain-free death. But if I was asked to, I'd refuse, even if it meant giving the condemned a less painful death. Because if such a protocol existed, if the legal system as a whole recognized nitrogen hypoxia as a quick and painless method of execution, that then removes a hurdle for pro-death-penalty states to start executing people.
I suspect I'm not alone because the knowledge I have is far from uncommon. Ask any pilot or astronaut or person involved with breathing gas systems and they'll tell you the same thing. Yet, few if any seem to be stepping up to 'solve' the problem of a humane execution protocol.

Gives me hope for the future of humanity.