Gao–Vitale conjecture on intrinsic-volume decay

Let $K$ be a convex GB-compact set in a separable Hilbert space, and let $m_k(K)$ denote its associated sequence of quantities. The sequence is required to satisfy one of two alternatives:

Gao–Vitale conjecture. Either

$$\lim_{k \to \infty} m_k(K)>0,$$

or

$$m_k(K)=O(k^{-1/2})\quad (k\to\infty).$$

Equivalently, if $m_k(K)$ tends to zero, it cannot decay at a rate slower than $k^{-1/2}$. The paper states that this conjecture is disproved: GB-sets with order $\rho(K)>\frac{2}{3}$ provide counterexamples.