The $beta$-delayed neutron emission probabilities of neutron rich Hg and Tl nuclei have been measured together with $beta$-decay half-lives for 20 isotopes of Au, Hg, Tl, Pb and Bi in the mass region N$gtrsim$126. These are the heaviest species where neutron emission has been observed so far. These measurements provide key information to evaluate the performance of nuclear microscopic and phenomenological models in reproducing the high-energy part of the $beta$-decay strength distribution. In doing so, it provides important constraints to global theoretical models currently used in $r$-process nucleosynthesis.
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