The case $d=3$ conjecture for weak and strong extremals of quasiconvex quadratic forms

Let ${\cal C}_3$ be the cone of $3\times 3$ quasiconvex quadratic forms, let a weak extremal mean an element that cannot be written as a sum of two non-proportional elements of ${\cal C}_3$ modulo null Lagrangians, and let an extreme ray be a one-dimensional face of ${\cal C}_3$. For $f\in{\cal C}_3$, let $\det{\mathop{\rm B}(\mathbf y)}$ denote the determinant of its acoustic tensor.

The case $d=3$ conjecture. Any non-polyconvex weak extremal $f\in {\cal C}_3$ is an extreme ray of ${\cal C}_3$. Moreover, if $f\in {\cal C}_3$ is a non-polyconvex extreme ray of ${\cal C}_3$, then $\det{\mathop{\rm B}(\mathbf y)}$ is an extremal polynomial different from a perfect square.

The conjecture proposes that, in dimension three, non-polyconvex weak extremality coincides with extremality of the cone, and that the determinant of the acoustic tensor of every non-polyconvex extreme ray is an extremal polynomial but not a perfect square. The surrounding discussion notes that the perfect-square case is expected to force polyconvexity, but no proof is known.