Morrey's rank-one convexity conjecture in two dimensions

Let $\mathbf E:\mathbb R^{m\times n}\rightarrow\mathbb R$ be a continuous function. It is rank-one convex when, for every $A\in\mathbb R^{m\times n}$ and every rank-one matrix $X$, the function $t\mapsto\mathbf E(A+tX)$ is convex. It is quasiconvex when

$$\int_{\mathbb R^n}[\mathbf E(A+D\eta)-\mathbf E(A)]\geqslant0$$

for every $A\in\mathbb R^{m\times n}$ and every $\eta\in\mathscr C^\infty_0(\mathbb R^n,\mathbb R^m)$.

Morrey's conjecture. Every rank-one convex function $\mathbf E:\mathbb R^{2\times2}\rightarrow\mathbb R$ is quasiconvex.

This is the two-dimensional case of the rank-one convexity versus quasiconvexity problem. The source presents it as an enduring open problem motivated by Morrey's theory and the known higher-dimensional counterexamples.