Actually this is true whichever interpretation you take, give or take some knowledge around black holes.
I think hawking actually proved that this is true regardless of how black holes work due to hawking radiation.
In the Copenhagen interpretation the collapse of the wave function explicitly violates unitarity (and thus reversibility).
I assume the following happens: while a (small) subsystem is in "pure state" (in quantum coherence), no information flows out of this subsystem. Then, when measuring, information flows out and other information flows in, which disturbs the pure state. This collapses of the wave function (quantum decoherence). For all practical purposes, it looks like quantum decoherence is irreversible, but technically this could still be reversible; it's just that the subsystem (that is in coherence) got much, much larger. Sure, for all practical purposes it's then irreversible, but for us most of physics anyway looks irreversible (eg. black holes).
But don't think that necessarily means we are in a Many-Worlds. I rather think that we don't have enough knowledge in this area. Assuming we live in a simulation, an alternative explanation would be, that unlikely branches are not further simulated to save energy. And in this case, superposition is just branch prediction :-)