Complete saturation and reduction existence conjecture for bodies

Let $K$ be a body in Euclidean space $cmathbb{E}^d$ or hyperbolic space $cmathbb{H}^d$. A packing is completely saturated if it is $n$-saturated for every $n\geq 1$, and a covering is completely reduced if it is $n$-reduced for every $n\geq 1$.

Complete saturation and reduction conjecture. Every body $K$ in $\mathbb{E}^d$ (respectively, in $\mathbb{H}^d$) admits a completely saturated packing and a completely reduced covering with replicas of $K$.

This proposes completely saturated packings and completely reduced coverings as local substitutes for optimal-density arrangements. The supplied text does not state whether the assertion has been proved or disproved.