Existence of an extremizer for the proposed Mahler-volume lower bound

Let $n\geq 4$, let $K$ be a symmetric convex set in $\mathbb{R}^n$, let $K^{\circ}$ denote its polar body, and let $\alpha(n)$ be the constant defined earlier in the source. The $n$-dimensional volume is denoted by $\operatorname{vol}_n$, and $\Gamma$ is the Gamma function.

Existence conjecture. There is a symmetric convex set $K$ in $\mathbb{R}^n$ such that

$$\operatorname{vol}_n(K)\operatorname{vol}_n(K^{\circ})=\frac{4\alpha(n-1)\pi^n}{\Gamma\left(\frac{n}{2}\right)^2}.$$

This conjecture asks whether the lower bound supplied by the paper's theorem is attained in every dimension $n\geq 4$. The source does not provide a resolution, and the notation $\alpha(n)$ is defined elsewhere in the paper.