  We study quasiconvex quadratic forms on $n \times m$ matrices which
correspond to nonnegative biquadratic forms in $(n,m)$ variables. We disprove a
conjecture stated by Harutyunyan--Milton (Comm. Pure Appl. Math. 70(11), 2017)
as well as Harutyunyan--Hovsepyan (Arch. Ration. Mech. Anal. 244, 2022) that
extremality in the cone of quasiconvex quadratic forms on $3\times 3$ matrices
can follow only from the extremality of the determinant of its acoustic tensor,
using previous work by Buckley--\v{S}ivic (Linear Algebra Appl. 598, 2020). Our
main result is to establish a conjecture of Harutyunyan--Milton (Comm. Pure
Appl. Math. 70(11), 2017) that weak extremal quasiconvex quadratics on $3
\times 3$ matrices are strong extremal. Our main technical ingredient is a
generalization of the work of Kunert--Scheiderer on extreme nonnegative ternary
sextics (Trans. Amer. Math. Soc. 370(6), 2018). Specifically, we show that a
nonnegative ternary sextic, which is not a square, is extremal if and only if
its variety (over the complex numbers) is a rational curve and all its
singularities are real.
