Conjecture that non-extremal acoustic determinant implies a non-extremal quadratic form

Let $f(\boldsymbol{\xi})=\boldsymbol{\xi}C\boldsymbol{\xi}^T$ be a quasiconvex quadratic form, where $\boldsymbol{\xi}\in\mathbb R^{3\times 3}$ and $C\in(\mathbb R^3)^4$ is a fourth-order tensor. Let $f(x,y)$ denote the associated form whose $y$-matrix has a determinant. Non-extremality conjecture. If the determinant of the $y$-matrix of $f(x,y)$ is not an extremal polynomial, then $f$ is not an extremal form.

This conjecture concerns the remaining case in the proposed characterization of $3\times3$ extremal quasiconvex quadratic forms. The preceding results treat cases where the determinant is irreducible, identically zero, or a perfect square; the conjecture asserts that the remaining case yields no extremal forms.