Milman–Szarek geometric lemma for easily covered polytopes

Let $r\in(0,1]$. A polytope $P=\operatorname{conv}(V)\subset\mathbb{R}^n$ contains $rB_2^n$ when $rB_2^n\subset P$, and $N(P,B_2^n)$ denotes the covering number of $P$ by translates of the Euclidean unit ball. Milman–Szarek conjecture. There exists $c=c(r)$ such that for every $n\in\mathbb{N}$ and every polytope $P=\operatorname{conv}(V)\subset\mathbb{R}^n$ containing $rB_2^n$,

$$N(P,B_2^n)<e^{cn}\implies |V|>e^{cn}.$$

In particular, the covering number and the number of vertices cannot both be subexponential in the dimension. Milman and Szarek showed that this conjecture implies the duality conjecture concerning covering numbers; although that duality conjecture has since been proved, the present conjecture remains open.