Exponential-facet threshold conjecture for volume approximation of the Euclidean ball

Let $D_n$ be the Euclidean unit ball in $\mathbb{R}^n$, let $N$ be a positive integer, and let $P^b_{n,N}$ be a polytope with at most $N$ facets that is best-approximating for $D_n$ with respect to the symmetric volume difference $\Delta_v$. Exponential-threshold conjecture. If $N\leq 2^n$ and the dimension is sufficiently large, then

$$\lim_{n\to\infty}\frac{\Delta_v(D_n,P^b_{n,N})}{|D_n|}=1.$$

This is proposed as a strengthening of the elementary result that the same limit equals $1$ when the number of facets is $e^{o(n)}$, motivated by symmetry considerations.