Collect data, and show how it’s unlilely unless your hypothesis is true.
The quantum immortality experiment doesn't do that, though. The outcome, by definition, always occurs within the realm of random chance. Your environment needs to create an outcome that is extremely unlikely to occur by random chance. The experiment is not repeatable. It makes no predictions about what's going to happen if you try again. It doesn't do anything useful to bolster the many worlds theory.
The experiment, as defined, only leads to your survival by random chance. The experiment does not create any outcome except by random chance so it cannot be used to prove anything.