Iwaniec's $L^p$ norm conjecture for the Ahlfors–Beurling operator

Let $T$ be the Ahlfors–Beurling operator on $L^p(\mathbb C)$, let $p^*=\max\{p,p/(p-1)\}$, and write $\|T\|_p$ for its operator norm. Iwaniec's conjecture. For every $p\in(1,\infty)$,

$$\|T\|_p=p^*-1.$$

The text describes this as an open problem originating with Iwaniec in 1982; it would imply optimal Sobolev integrability for quasiconformal maps and has further consequences in the calculus of variations.