Conjugator length in Thompson's groups

Abstract: We prove Thompson's group $F$ has quadratic conjugator length function. That is, for any two conjugate elements of $F$ of length $n$ or less, there exists an element of $F$ of length $O(n^2)$ that conjugates one to the other. Moreover, there exist conjugate pairs of elements of $F$ of length at most $n$ such that the shortest conjugator between them has length $\Omega(n^2)$. This latter statement holds for $T$ and $V$ as well.