The claim has interesting ramifications but the hypothesis cannot be proven nor disproven.

First, assume we have a “quantum guillotine” – a gadget that ends a consciousness based on quantum phenomena.
My understanding of the claim is, that if someone uses the gadget, then they would perceive that the gadget always fails because their consciousness “jumps” over to the universe where it lives on.
Their world would be quite weird because even after many repetitions the result would be the same.
After one hundred repetitions the probability that the claim works is practically certain.
However, at the same time the proportion of the universes where the experiment always fails decreases exponentially.

Observers (other than the one in question) would have a very different experience of the experiment.
They would really see a random phenomena – 50% of the time the gadget would turn on, 50% it wouldn’t.
This is equivalent to the results when the claim does not work, so there is no way for them to distinguish the two.
The probability of seeing 100 successful experiments performed by someone else is extremely low so in a universe where this happens the people would be all excited, but they still would not be able to differentiate it from a random event.

Interestingly enough, the performer would also be unable to prove whether the claim works or not.
The many worlds interpretation splits consciousness as well so killing only one of them lets the other one alive.
In this view, the experiment kills in 50% of the universes and one could not say in which one of them he will end up.
There, the one branch that prevailed is tied to the quantum experiment.
So one lives in exactly the universes where the quantum experiment happened in a certain way.

If many world hypothesis is not true, then performing the experiment has 50% chance of a bad outcome and that is all.