Uniformly separated bounded sets are $ell^1$-bounded

Let $H$ be the Hilbert space under consideration, and let $M\subseteq H$ be bounded. The set $M$ is uniformly separated if

$$\inf_{x\ne y\in M}\|x-y\|\ne 0.$$

Uniform separation conjecture. If $M$ is bounded and uniformly separated, then $M$ is $ell^1$-bounded. This is posed as an open question in the paper; no resolution is supplied there.