(emphasis added)

PIEZO WARNING:
A lighter with a piezo-electric ignition can be a great thing, but they are unreliable at altitudes above about 5,000' and typically useless above 10,000. As I recall the White Mountains in AZ are quite high.

I was taught the above in a Mountaineering class. I can personally vouch for not being able to use my Windmill piezo-electric at 8000'.