The geometric lemma for convex hulls and mean width

Let $D$ denote the Euclidean unit ball in $\mathbb{R}^n$, and let $M^*(A)$ denote the mean width functional used in the paper. For a set $S=\{x_i\}_{i=1}^{2^k}\subset\mathbb{R}^n$, put $K=\operatorname{conv}(S)$ and assume $N(K,D)\le 2^k$. Geometric lemma. There exists an absolute constant $C_0$ such that, for every dimension $n$ and every such set $S$,

$$M^*(K\cap D)\le C_0\sqrt{\frac{k}{n}}.$$

The lemma is introduced as a conjectural ingredient for improving the exponent in the paper's main entropy-duality theorem. Its status is not resolved by the supplied text.