Coarse embeddings into $c_0(\mathrm{\Gamma })$

Let $\lambda$ be a large enough cardinal number (assuming GCH it suffices to let $\lambda ={\mathrm{\aleph }}_{\omega }$). If $X$ is a Banach space with $\text{dens}(X)\ge \lambda$, which admits a coarse (or uniform) embedding into any $c_0(\mathrm{\Gamma })$, then $X$ fails to have nontrivial cotype, i.e. $X$ contains ${\mathrm{\ell }}_{\mathrm{\infty }}^n$ $C$-uniformly for every $C>1$. In the special case when $X$ has a symmetric basis, we may even conclude that it is linearly isomorphic with $c_0(\text{dens}X)$.