Concerning the problem of extremality of quasiconf'ormal mappings withdilatation bounds. we discuss the unique extremality of the problem and prove the if part of aconjecture on the unique extremality ([GI], [RI]). To this end. we need to investigate a newextremal problem in the infinitesimal setting. In particular, we give a complete description ofthe unique infinitesimal extremality of partially zero Beltrami differentials.