Conjugator Length in the Baumslag-Gersten Group

We show that the conjugator length function of the Baumslag-Gersten group is bounded above and below by a tower of exponentials of logarithmic height – in particular it grows faster than any tower of exponentials of fixed height. We conjecture that no one-relator group has a larger conjugator length function than the Baumslag-Gersten group. Along the way, we also show that the conjugator length function of the $m$-th iterated Baumslag-Solitar groups is equivalent to the $m$-times iterated exponential function.