Iwaniec's quasiconvexity conjecture for the planar Burkholder integrand

Let $B_p\colon \mathbb{R}^{2\times 2}\to\mathbb{R}$ be the Burkholder integrand, defined by

$$B_p(A)=\left(\left|1-\frac{2}{p}\right|\lvert A\rvert^2+\det A\right)\lvert A\rvert^{p-2}, \qquad p\geq 1,$$

where $\lvert A\rvert$ denotes the operator norm. Iwaniec's conjecture. The integrand $B_p\colon \mathbb{R}^{2\times 2}\to\mathbb{R}$ is quasiconvex. This is a long-standing open problem concerning whether rank-one convexity implies quasiconvexity for the isotropic, positively homogeneous Burkholder integrand in two dimensions.