Sharpness conjecture for the Hankel–Korn inequality

Let $d$ be the dimension, let $cmathcal H(d)$ denote the Hankel subspace, and let $p^*=\max\{p,p/(p-1)\}$ be the conjugate exponent. Write $C(\mathcal H(d),p)$ for the optimal $L^p$ Hankel–Korn constant and $C^{\mathrm{rc}}(\mathcal H(d),p)$ for the corresponding rank-one-convex constant. Sharpness conjecture. One has

$$C(\mathcal H(d),p)=C^{\mathrm{rc}}(\mathcal H(d),p)=C(\mathcal H(d),2)(p^*-1).$$

The preceding bounds establish the upper estimate for the rank-one-convex constant and the lower estimate $C^{\mathrm{rc}}(\mathcal H(d),p)\geq p^*-1$; the conjecture asserts that these bounds are sharp and that the ordinary and rank-one-convex constants coincide. Its resolution would determine the precise $L^p$ dependence of the Hankel–Korn inequality.