Quasiconvexity conjecture for the Korn variational integrand

Let $d$ be the dimension, let $P_{\mathrm{Skew}(d)}$ and $P_{\mathrm{Sym}(d)}$ be the orthogonal projections onto the skew-symmetric and symmetric matrices, and let $\mathcal G_p$ be the function introduced in the paper. For $A\in M_d(\mathbb R)$, consider

$$-\mathcal G_p\bigl(|P_{\mathrm{Skew}(d)}(A)|,|P_{\mathrm{Sym}(d)}(A)|\bigr).$$

Quasiconvexity conjecture. The displayed function is quasiconvex at some matrix $A_0$. By homogeneity, quasiconvexity at $A_0$ implies quasiconvexity at $0$. This proposed strengthening concerns the variational structure behind the sharp Korn constant and links the analytic inequality to quasiconvexity; the supplied excerpt does not state whether it has been resolved.