Strong isotropic constant conjecture for symmetric convex bodies

Let $\mathcal{C}(K)=\lvert K\rvert^2/\det\operatorname{Cov}(K)=1/L_K^{2n}$ for a convex body $K\subseteq\mathbb{R}^n$. The strong isotropic constant conjecture for symmetric convex bodies. For every symmetric convex body $K\subseteq\mathbb{R}^n$,

$$\mathcal{C}(K)\geq\mathcal{C}([-1,1]^n)=12^n.$$

Equivalently, the isotropic constant is maximized by the cube among symmetric convex bodies. The conjecture is presented as the symmetric case of a strong isotropic conjecture, with no resolution supplied here.