I'm not sure that's how it works. As I understand it, the model predicts bubble nucleation, not state leakage, due to spatial inhomogeneities: a local energy fluctuation leads to a true-vacuum bubble forming and expanding.

However, let's leave aside the spatial inhomogeneity for the moment. As I understand it, tunneling in many worlds would result in a continuum of decayed worlds being continuously spawned, all with equal and infinitesimal probability, with the un-decayed one slowly decreasing in probability.

Assuming a Schrodinger cat-type experiment, with Eliezer being the cat, and assuming that Eliezer dies in every decayed world (not an unreasonable assumption if it's vacuum that decays), and assuming quantum immortality-type ontology (quite a number of assumptions), Eliezer will only ever perceive the surviving branch, with no measurable leakage.