Euclidean ball extremality conjecture for finite-facet volume approximation

Fix $n\in\mathbb{N}$ with $n\geq 2$ and $N\geq n+1$, and let $K$ be a convex body in $\mathbb{R}^n$. For a polytope $P$ with at most $N$ facets, write $\Delta_v(P,K)$ for the symmetric volume difference and $|K|$ for the volume of $K$. Euclidean-ball extremality conjecture. Then

$$\min_{P\text{ has at most }N\text{ facets}}\frac{\Delta_v(P,K)}{|K|}\leq\min_{P\text{ has at most }N\text{ facets}}\frac{\Delta_v(P,D_n)}{|D_n|}.$$

The conjecture asserts that, among convex bodies normalized against their own volume, the Euclidean ball is hardest to approximate by polytopes with at most $N$ facets. It is motivated by the affine isoperimetric inequality, which gives the corresponding extremal statement for the leading asymptotic constant as $N\to\infty$; the finite-$N$ inequality remains open.