In work dating to 1981, two different sufficient conditions for unique extremality of quasiconformal mappings, referred to here as Theorem~A and Theorem~B were found. In the case of Theorem~A, the condition is now known to also be necessary. In spite of this, Theorem~B has been found to be the more useful of the two conditions in delicate situations. However, the question of whether the conditions of Theorem~B are necessary is still open. We discuss some matters relevant to this problem. As a new illustration of Theorem~B, we show that it can be used to give a self-contained proof of a result of Strebel's.