The six-vertex Mahler-type conjecture for centrally symmetric planar convex bodies

Let $B\in\mathcal{K}^2_o$ be an $o$-symmetric convex body in $\mathbb{R}^2$, let $Q_6(B)$ denote the associated six-vertex convex polygon, and let $B^{\circ}$ be the polar body of $B$. Six-vertex Mahler-type conjecture.

$$\lambda_2(Q_6(B))\lambda_2(B^{\circ})\geq 8,$$

with equality if $B$ is a parallelogram. This conjecture would imply that $m^{HT}(n,2)=\frac{8}{\pi}$ for all $n\geq 7$; the surrounding discussion presents it as an open question motivated by Mahler's planar volume-product inequality.