The mixed-radius Lee sphere tiling conjecture

A Lee sphere in $\mathbb{R}^n$ is a cross-polytope-type region defined by a Lee radius, and a tiling means a covering of $\mathbb{R}^n$ by translates of such spheres with disjoint interiors. Mixed-radius Lee sphere tiling conjecture. For $n\geq 3$, there does not exist a tiling of $\mathbb{R}^n$ with Lee spheres of radius at least $1$ such that the radius of at least one of them is at least $2$. The paper presents this as a strengthening of the Golomb--Welch conjecture; it notes that the corresponding assertion with all radii at least two is known in the relevant dimensions, while this stronger mixed-radius statement remains open.