Does wrapping banana stems actually prolong their life?
Revised answer (old answer removed due to accidentally swerving off-topic and to some extent containing incorrect information)

Even though ethylene biosynthesis happens throughout the banana most-likely none of it occurs in the stem itself as it contains a small percentage of the metabolically active tissue. There aren't as many studies that focus on the stem of the banana specifically as I imagine it's not a very attractive topic, but the current-day findings that we have highly support the assumption of insignificant ethylene gas production (or none at all) in the banana stem and it's why wrapping it will pose virtually no difference in the further ripening of a banana.

Links to relevant studies: (1) (2) (3)