Waiting times for particles in a branching Brownian motion to reach the rightmost position

It has been proved by Lalley and Sellke (1987) [13] that every particle born in a branching Brownian motion has a descendant reaching the rightmost position at some future time. The main goal of the present paper is to estimate asymptotically as ss goes to infinity, the first time that every particle alive at the time ss has a descendant reaching the rightmost position.