The approximation-radius gap conjecture for centrally symmetric convex bodies

Let $X\subseteq\mathbb{R}^n$ be a centrally symmetric convex body, let $R(X,\sigma)$ denote its minimax risk, and let $z_{c,k}(X)$ denote the approximation-radius quantity defined in the paper for $0<c\leq 1$ and dimension $k$. The approximation-radius gap conjecture. For every $0<c\leq 1$, there exists a constant $f(c)>0$ such that

$$R(X,\sigma)\leq\sqrt{\log n}\,f(c)\max_k\min\bigl(z_{c,k}(X)^2,k\sigma^2\bigr).$$

The conjecture would show that the approximation-radius lower bound loses at most a factor of order $\sqrt{\log n}$; the authors motivate it from the known gap for $\ell_p$ balls with $1\leq p<2$, while the supplied text gives no resolution.