Conjugator length in Thompson's groups

We prove that Thompson's group (Formula presented.) has quadratic conjugator length function. That is, for any two conjugate elements of (Formula presented.) of length (Formula presented.) or less, there exists an element of (Formula presented.) of length (Formula presented.) that conjugates one to the other. Moreover, there exist conjugate pairs of elements of (Formula presented.) of length at most (Formula presented.) such that the shortest conjugator between them has length (Formula presented.). This latter statement holds for (Formula presented.) and (Formula presented.) as well.